chapter 1 - geometry 2

Arc: Part of a curve, most commonly a portion of the distance around the circumference of a circle

Chord: A straight line joining two points on the circumference of a circle

Concyclic points: Points that lie on the circumference of the same circle

Congruent: Two figures are congruent if they have the same size and shape. They are identical in every way

Cyclic quadrilateral: A cyclic quadrilateral is a figure whose four vertices are concyclic points. The four vertices lie on the circumference of a circle

Polygon: A polygon is a closed plane figure with straight sides

Radius: A radius is the distance from the centre of a circle out to the circumference (radii is plural, meaning more than one radius)

Similar: Two figures are similar if they have the same shape but a different size. Corresponding angles are equal and corresponding sides are in the same ratio

Subtend: Form an angle at some point (usually the centre or circumference)

Tangent: A straight line external to a circle that just touches the circle at a single point

Vertex: A vertex is a corner of a figure (vertices is plural, meaning more than one vertex)

Terminology

Geometry 2

1

3Chapter 1 Geometry 2

inTroduCTion

You studied geometrY in the Preliminary course. in this chapter, you will revise this work and extend it to include some more general applications of geometrical properties involving polygons in 2 unit and circles in extension 1.

You will also use the Preliminary topic on straight-line graphs to explore coordinate methods in geometry.

Plane Figure Geometry

Here is a summary of the geometry you studied in the Preliminary course.

Vertically opposite angles

AEC DEBand+ + are called vertically opposite angles. AED CEBand+ + are also vertically opposite angles.

Vertically opposite angles are equal.

Parallel lines

if the lines are parallel, then alternate angles are equal.

if the lines are parallel, then corresponding angles are equal.

4 maths in Focus Mathematics Extension 1 HSC Course

if the lines are parallel, cointerior angles are supplementary (i.e. their sum is 180°).

TesTs for parallel lines

if alternate angles are equal, then the lines are parallel.

if corresponding angles are equal, then the lines are parallel.

if cointerior angles are supplementary, then the lines are parallel.

Angle sum of a triangle

the sum of the interior angles in any triangle is 180°, that is, a b c 801+ + =

If ,AEF EFD+ += then AB || CD.

If ,BEF DFG+ += then AB || CD.

If ,BEF DFE 180c+ ++ = then AB || CD.


Exterior angle of a triangle

the exterior angle in any triangle is equal to the sum of the two opposite interior angles. that is,

x y z+ =

Congruent triangles

two triangles are congruent if they are the same shape and size. All pairs of corresponding sides and angles are equal.

For example:

We write .ABC XYZ/∆ ∆

TesTs

to prove that two triangles are congruent, we only need to prove that certain combinations of sides or angles are equal.

two triangles are congruent if• SSS: all three pairs of corresponding sides are equal• SAS: two pairs of corresponding sides and their included angles are equal• AAS: two pairs of angles and one pair of corresponding sides are equal• RHS: both have a right angle, their hypotenuses are equal and one

other pair of corresponding sides are equal

Similar triangles

triangles, for example ABC and XYZ, are similar if they are the same shape but different sizes.

As in the example, all three pairs of corresponding angles are equal.All three pairs of corresponding sides are in proportion (in the same ratio).

The included angle is the angle between the 2 sides.


This shows that all 3 pairs of sides are in proportion.

two triangles are similar if:three pairs of • corresponding angles are equalthree pairs of • corresponding sides are in proportiontwo pairs of • sides are in proportion and their included angles are equal

Ratios of intercepts

When two (or more) transversals cut a series of parallel lines, the ratios of their intercepts are equal.

that is, : :AB BC DE EF=

or BCAB

EFDE=

Pythagoras’ theorem

the square on the hypotenuse in any right angled triangle is equal to the sum of the squares on the other two sides.

that is, c a b2 2 2= +or c a b2 2= +

We write: XYZ∆;ABC <∆XYZ∆ is three times larger than ABC∆

ABXY

ACXZ

BCYZ

26 3

412 3

515 3

= =

= =

= =

ABXY

ACXZ

BCYZ

` = =

TesTs

there are three tests for similar triangles.

If 2 pairs of angles are equal then the third pair must also be equal.


Quadrilaterals

A quadrilateral is any four-sided figure

in any quadrilateral the sum of the interior angles is 360°

parallelogram

A parallelogram is a quadrilateral with opposite sides parallel

Properties of a parallelogram:• opposite sides of a parallelogram are equal• opposite angles of a parallelogram are equal• diagonals in a parallelogram bisect each other

each diagonal bisects the parallelogram into two • congruent triangles

TesTsA quadrilateral is a parallelogram if:

both pairs of • opposite sides are equalboth pairs of • opposite angles are equal

• one pair of sides is both equal and parallelthe • diagonals bisect each other

These properties can all be proved.


rhombus

A rectangle is a parallelogram with one angle a right angle

Properties of a rectangle:the same as for a parallelogram, and also•diagonals are equal•

TesT

A quadrilateral is a rectangle if its diagonals are equal

Application

Builders use the property of equal diagonals to check if a rectangle is accurate. For example, a timber frame may look rectangular, but may be slightly slanting. Checking the diagonals makes sure that a building does not end up like the Leaning Tower of Pisa!

A rhombus is a parallelogram with a pair of adjacent sides equal

Properties of a rhombus:the same as for parallelogram, and also•diagonals bisect at right angles•diagonals bisect the angles of the rhombus•

It can be proved that all sides are equal.

If one angle is a right angle, then you can prove all angles are right angles.

recTangle


TesTs

A quadrilateral is a rhombus if:all sides are equal•diagonals bisect each other at right angles•

square

A square is a rectangle with a pair of adjacent sides equal

Properties of a square:the same as for rectangle, and also•diagonals are perpendicular•diagonals make angles of 45° with the sides•

Trapezium

A trapezium is a quadrilateral with one pair of sides parallel

KiTe

A kite is a quadrilateral with two pairs of adjacent sides equal


square

the sum of the exterior angles of any polygon is 360°

Areas

most areas of plane figures come from the area of a rectangle.

recTangle

A lb=

Triangle

A x2=

A bh21=

A square is a special rectangle.

The area of a triangle is half the area of a rectangle.

A polygon is a plane figure with straight sides

A regular polygon has all sides and all interior angles equal

the sum of the interior angles of an n-sided polygon is given by

°( )S n 2 180= −

Polygons

´


parallelogram

A bh=

rhombus

A xy21=

(x and y are lengths of diagonals)

Trapezium

( )A h a b21= +

circle

πA r2=

the following examples and exercises use these results to prove properties of plane figures.

You will study the circle in more detail. See Chapter 5.

The area of a parallelogram is the same as the area of two triangles.


ExAmplEs

1. Prove A C+ += in parallelogram ABCD.

Solution

°

° °

° ° °° ° °

°

Let

Then ( , cointerior angles, )

( ) ( , cointerior angles, )

A x

B x A B AD BC

C x B C AB DCx

x

A C

180

180 180180 180

`

+

+ + +

+ + +

+ +

<

<

== −= − −= − +==

2. triangle ABC below is isosceles with AB AC= . Prove that the base angles of ABC∆ are equal by showing that ABD∆ and ACD∆ are congruent.

Solution

° ( )

( )

ADB ADC

AB AC

90 given

given

+ += ==

AD is common ABD ACD/∆ ∆ (rHs)

so ABD ACD+ += (corresponding angles in congruent )s∆ base angles are equal

\

\


3. Prove that opposite sides in a parallelogram are equal.

Solution

Let ABCD be any parallelogram and draw in diagonal AC.

( , )

( , )

DAC ACB AD BC

BAC ACD AB DC

alternate s

alternate s

+ + +

+ + +

<

<

==

AC is common.` ABC ADC/∆ ∆ (AAs)

( )

( )

AB DC

AD BC

corresponding sides in congruent s

similarly

` ∆==

` opposite sides in a parallelogram are equal

1.

DE bisects acute angle ABC+ so that .ABD CBD+ += Prove that DE also bisects reflex angle .ABC+ that is, prove

.ABE CBE+ +=

2.

Prove that CD bisects .AFE+

3. Prove .VW XY<

4.

given °,x y 180+ = prove that ABCD is a parallelogram.

1.1 exercises


5. BD bisects .ABC+ Prove that .ABD CBD/∆ ∆

6. (a) show that .ABC AED/∆ ∆Hence prove that (b) ACD∆ is

isosceles.

7. ABCD is a square. Lines are drawn from C to M and N, the midpoints of AD and AB respectively. Prove that .MC NC=

8. OC is drawn so that it is perpendicular to chord AB and o is the centre of the circle. Prove that OAC∆ and OBC∆ are congruent, and hence that OC bisects AB.

9. CE and BD are altitudes of ,ABC∆ and ABC∆ is isosceles ( ).AB AC= Prove that .CE BD=

10. ABCD is a kite where AB AD= and .BC DC= Prove that diagonal AC bisects both DAB+ and

.DCB+

11. MNOP is a rhombus with .MN NO=

show that(a) MNO∆ is congruent to MPO∆(b) PMQ NMQ+ +=(c) PMQ∆ is congruent to NMQ∆(d) °MQN 90+ =

12. show that a diagonal cuts a parallelogram into two congruent triangles.

13. Prove that opposite angles are equal in any parallelogram.

The altitude is perpendicular to the other side of the triangle


14. ABCD is a parallelogram with .BM DN= Prove that AMCN is

also a parallelogram.

15. ABCD and BCEF are parallelograms. show that AFED is a parallelogram.

16. ABCD is a parallelogram with .DE DC= Prove that CE bisects

.BCD+

17. in quadrilateral ABCD, AB CD= and .BAC DCA+ += Prove ABCD is a parallelogram.

18. ABCD is a parallelogram with .°AEB 90+ = Prove

(a) AB BC=(b) ABE CBE+ +=

19. Prove that the diagonals in any rectangle are equal.

20. Prove that if one angle in a rectangle is 90° then all the angles are 90°.

21. ABCD is a rhombus with .AD CD= Prove that all sides of

the rhombus are equal.

22. ABCD is an isosceles trapezium. Prove the base angles ADC+ and

BCD+ are equal.

23. Prove that ADC ABC+ += in kite ABCD.

24. in rectangle ABCD, E is the midpoint of CD. Prove .AE BE=


25. ABCD is a rhombus.Prove (a) ADB∆ and BCD∆ are

congruent.Hence show (b) .ABE CBE+ +=Prove (c) ABE∆ and CBE∆ are

congruent.Prove (d) .°AEB 90+ =

Surface Areas and Volumes

Areas are used in finding the surface area and volume of solids. Here is a summary of some of the most common ones.

SURFACE AREA VOLUME

( )S lb bh lh2= + + V lbh=

S x6 2= V x3=

( )πS r r h2= + πV r h2=

πS r4 2= πV r34 3=

You will need some of these formulae when you study maxima and minima problems in Chapter 2.


( )πS r r l= + πV r h31 2=

in general, the volume of any prism is given byV Ah=

where A is the area of the base and h is its height

in general, the volume of any pyramid is given by

V Ah31=

Where A is the area of the base and h is its height


While surface area and volume is not a part of the geometry in the Hsc syllabus, the topic in chapter 2 uses calculus to find maximum or minimum areas, perimeters, surface areas or volumes. so you will need to know these formulae in order to answer the questions in the next chapter. Here are some questions to get you started.

ExAmplE

Find the surface area of a cone whose height is twice the radius, in terms of r.

Solution

`

( )

h r

l r h

r r

r r

r

l r

r

2

2

4

5

5

5

2 2 2

2 2

2 2

2

2

== += += +=

==

surface area ( )πS r r l= + where l is slant height ( )πr r r5= +

1. A rectangular prism has dimensions 12.5 mm, 84 mm and 64 mm. Find its

surface area and(a) volume.(b)

2. A sphere has a volume of 120π m3. Find the exact value of r.

3. A rectangular prism has dimensions x, x + 2 and 2x – 1. Find its volume in terms of x.

1.2 exercises


4. A cylinder has a volume of 250 cm3. if its base has radius r and its

height is h, show that πh

r 250= .

5. Find the volume of a cylinder in terms of r if its height is five times the size of its radius.

6. the ratio of the length to the breadth of a certain rectangle is 3:2. if the breadth of this rectangle is b units, find a formula for the area of the rectangle in terms of b.

7. Find the volume of a cube with sides ( ) .x 2 cm+

8. What would the surface area of a cylinder be in terms of h if its height is a third of its radius?

9. A square piece of metal with sides 3 m has a square of side x cut out of each corner. the metal is then folded up to form a rectangular prism. Find its volume in terms of x.

10. A cone-shaped vessel has a height of twice its radius. if i fill the vessel with water to a depth of 10 cm, find the volume of water to the nearest cm3.

11. the area of the base of a prism is given by 3h 2, where h is the height of the prism. Find a formula for the volume of the prism.

12. the area of the base of a pyramid is 6h 15 where h is the height of the pyramid. Find the volume of the pyramid in terms of h.

13. A rectangular pyramid has base dimensions x – 3 and 3x 5, and a height of 2x 1. Write a formula for the volume of the pyramid in terms of x.

14. the height of a rectangular prism is twice the length of its base. if the width of the base is x and the length is 3x – 1, find an expression for the

volume and(a) surface area of the prism.(b)

15. Find a formula for the slant height of a cone in terms of its radius r and height h.

16. A page measuring x by y is curved around to make an open cylinder with height y. Find the volume of the cylinder in terms of x and y.

17. the volume of a cylinder is 400 cm3. Find the height of the cylinder in terms of its radius r.

18. A cylinder has a surface area of 1500 cm2. Find a formula for its height h in terms of r.

19. the surface area of a cone is given by ( )πS r r l= + where l is the slant height. Find a formula for the slant height of a cone with surface area 850 cm2 in terms of r.

+

+

++


20. A rectangular timber post is cut out of a log with diameter d as shown. if the post has length x and breadth y, write y in terms of x when d 900 mm.

d

did you KnoW?

regulAr SolidS

There are only five solids with each face the same size and shape. These are called platonic solids. Research these on the internet.

=


The skeletons opposite are those of radiolarians. These are tiny sea animals, with their skeletons shaped like regular solids.

A salt crystal is a cube. A diamond crystal is an octahedron.

Diamond crystal

Coordinate Methods in Geometry

Problems in plane geometry can be solved by using the number plane.You studied straight-line graphs in the Preliminary course. some of the

main results that you learned will be used in this section. You may need to revise that work before studying this section.

Here is a summary of the main formulae.

Distance

the distance between two points ( , )x y1 1 and ( , )x y2 2 is given by

d x x y y2 12

2 12= − + −_ _i i

Midpoint

the midpoint of two points ( , )x y1 1 and ( , )x y2 2 is given by

,Px x y y

2 21 2 1 2=

+ +e o

Gradient

the gradient of the line between ( , )x y1 1 and ( , )x y2 2 is given by

m x xy y

2 1

2 1= −−

the gradient of a straight line is given byθtanm =

where θ is the angle the line makes with the x-axis in the positive direction.


the gradient of the line ax by c 0+ + = is given by

mba= −

Equation of a straight line

the equation of a straight line is given by( )y y m x x1 1− = −

where ( , )x y1 1 lies on the line with gradient m.

Parallel lines

if two lines are parallel, then they have the same gradient. that is,m m1 2=

Perpendicular lines

if two lines with gradients m1 and m2 respectively are perpendicular, then

m m m m1 1or1 2 21

= − = −

Perpendicular distance

the perpendicular distance from ( , )x y1 1 to the line ax by c 0+ + = is given by

| |d

a b

ax by c2 2

1 1=+

+ +

Ratios

the coordinates of a point P that divides the interval between points ( , )x y1 1 and ( , )x y2 2 in the ratio :m n respectively are given by

.x m nmx nx

y m nmy ny

and2 1 2 1= ++

= ++

if P divides the interval externally in the ratio :m n, then the ratio is negative.that is, : or : .m n m n− −

Angle between two lines

the acute angle θ between two straight lines is given by

θtanm m

m m

1 1 2

1 2=+

−

where m1 and m2 are the gradients of the lines.


ExAmplEs

1. show that triangle ABC is right angled, where ( , ), ( , )A B3 4 1 1= = − − and ( , )C 2 8= − .

Solution

method 1:

( ) ( )d x x y y2 12

2 12= − + −

( ) ( )( ) ( )

( ) ( )( )

( ( )) ( )( )

AB

AC

BC

1 3 1 4

4 5

16 25

41

2 3 8 4

5 4

25 16

41

1 2 1 8

1 9

1 81

82

2 2

2 2

2 2

2 2

2 2

2 2

= − − + − −= − + −= +== − − + −= − += +== − − − + − −= + −= +=

AB AC

BC

41 4182

2 2

2

+ = +==

since Pythagoras’ theorem is true, the triangle ABC is right angled.

method 2:

m x xy y

m

m

1 31 4

45

45

2 38 4

54

54

AB

AC

2 1

2 1= −−

=− −− −

=−−

=

=− −

−

=−

= −

ConTinued


m m45

54

1

AB AC = −

= −

so AB and AC are perpendicularso triangle ABC is right angled at A.

2. Prove that points , , ,A B1 1 2 1− −^ ^h h and ( , )C 4 3 are collinear.

Solution

collinear points lie on the same straight line, so they will have the same gradient.

m x xy y

2 1

2 1= −−

( )( )

m

m

m m

2 11 1

32

32

4 2

3 1

64

32

AB

BC

AB BC=

= − −− −

= −−

=

=− −− −

=

=

so the points are collinear.

1. show that points ( , ),A 1 0−( , ), ( , )B C0 4 7 0 and ( , )D 6 4− are

the vertices of a parallelogram.

2. Prove that ( , ), ( , )A B1 5 4 6− and ( , )C 3 2− − are vertices of a right

angled triangle.

3. given ABC∆ with vertices ( , ), ( , )A B0 8 3 0 and ( , )C 3 0−

show that (a) ABC∆ is isoscelesfind the length of the altitude (b)

from Afind the area of the triangle.(c)

4. show that the points , ,X 3 2^ h

( , )Y 2 1− and ( , )Z 8 3 are collinear.

5. (a) show that the points , ,A 2 5^ h

( , ), ( , )B C1 0 7 4− − and ( , )D 3 4− are the vertices of a kite.

Prove that the diagonals of (b) the kite are perpendicular.

show that (c) CE AE2= where E is the point of intersection of the diagonals.

6. Find the radius of the circle that has its centre at the origin and a tangent with equation given by

.x y4 3 5 0− − =

1.3 exercises

´


7. (a) Find the equation of the perpendicular bisector of the line joining ,( )A 3 2 and ,( ).B 1 8−

show that the point (b) ,( )C 7 9 lies on the perpendicular bisector.

What type of triangle is (c) ?ABC∆

8. show that OAB∆ and OCD∆ are similar where ( , ), ( , ), ( , )0 7 2 0 0 14− and ( , )4 0− are the points A, B, C and D respectively and O is the origin.

9. (a) Prove that OAB∆ and OCB∆ are congruent given ( , ), ( , ), ( , )A B C3 4 5 0 2 4− and O the

origin.show that (b) OABC is a

parallelogram.

10. the points ( , ), ( , ), ( , )A B C0 0 2 0 2 2 and ( , )D 0 2 are the vertices of a square. Prove that its diagonals make angles of 45° with the sides of the square.

11. Prove that ( , ), ( , ),P Q2 0 0 5− ( , )R 10 1 and ( , )S 8 4− are the

vertices of a rectangle.

12. the points ( , ), ( , )A B5 0 1 4− and ( , )C 3 0 form the vertices of a

triangle.Find (a) X and Y, the midpoints

of AB and AC respectively.show that (b) XY and BC are

parallel.show that (c) BC 2XY.

13. show that the diagonals of a square are perpendicular bisectors, given the vertices of square ABCD where

, , , , ,A a B a a C a0 0= − = − =^ ^ ^h h h and ( , )D 0 0= .

14. (a) show that points ,X 3 2^ h and ( , )Y 1 0− are equal distances from

the line .x y4 3 1 0− − =Find (b) Z, the x-intercept of the

line.What is the area of triangle (c)

XYZ?

15. ABCD is a quadrilateral with , , , , ,A B C3 1 1 4 5 2− − −^ ^ ^h h h

and , .D 4 3−^ h show that the midpoints of each side are the vertices of a parallelogram.

in this section you will use the properties from the Preliminary course to answer questions involving proofs.

on the next page is a summary of the properties. You may need to revise them before doing the exercises.

Circle Properties

=


Arcs and chords

1.equal arcs subtend equal angles at the centre of the circle.

the converse is also true:

if two arcs subtend equal angles at the centre of the circle, then the arcs are equal.

2.equal chords subtend equal angles at the centre of the circle.


equal angles subtended at the centre of the circle cut off equal chords.


Angle properties

1.

the angle at the centre of a circle is twice the angle at the circumference subtended by the same arc.

2.Angles in the same segment of a circle are equal.

3.the angle in a semicircle is a right angle.

Chord properties

1.A perpendicular line from the centre of a circle to a chord bisects the chord.



A line from the centre of a circle that bisects a chord is perpendicular to the chord.

2.equal chords are equidistant from the centre of the circle.


chords that are equidistant from the centre are equal.

Class Exercise

Prove that chords that are equidistant from the centre are equal.


3.the products of intercepts of intersecting chords are equal.

AE EB DE EC$ $=

Cyclic quadrilaterals

A cyclic quadrilateral is a figure whose 4 vertices are concyclic points.

1.the opposite angles in a cyclic quadrilateral are supplementary.


if the opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic.


2.the exterior angle at a vertex of a cyclic quadrilateral is equal to the interior opposite angle.

Tangent properties

1.the tangent to a circle is perpendicular to the radius drawn from the point of contact.


the line perpendicular to the radius at the point where it meets the circle is a tangent to the circle at that point.

2.tangents to a circle from an exterior point are equal.


3.When two circles touch, the line through their centres passes through their point of contact.

4.the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

5.the square of the length of the tangent from an external point is equal to the product of the intercepts of the secant passing through this point.

P

Q

RS

PQ QR QS2 $= where PQ is a tangent to the circle.


ExAmplEs

1.

given ,AC GE= prove that .BD FD=

Solution

`

`

°

Join

( )

is common

(equal chords equidistant from the centre)

by RHS

( sides in congruent )

OD

OFD OBD

OD

OF OB

OFD OBD

BD FD

90 given

corresponding s

+ +

/∆ ∆∆

= =

=

=

2. Prove .AB ED<

Solution

( equal to opposite int in cyclic quad)

( )

GED DCF

DCF BAF

GED BAF

ext

similarly

`

+ + + +

+ +

+ +

===

these are equal corresponding anglesAB ED` <


3. Prove .COA CAB2+ +=

O

C

B

A

Solution

Let

(tangents from external point equal)

is isosceles

(base s of isosceles )

(tangent radius)°

°

CAB x

AB CB

ABC

ACB x

OAB

OAC CAB OAB

OAC OAB CAB

x90

90

`

`

`

=

+

+ +

+

+ + +

+ + +

∆∆

==

==

+ == −= −

Similarly,

( ) (angle sum of )

°

° °

° °

OCA x

COA x

x

x

CAB

90

180 2 90

180 180 2

2

2

+

+

+

∆= −= − −= − +==

4. given ,AB DE< prove ABC∆ is isosceles.

Solution

( s in alternate segment)

But ( s, )

DCA ABC

DCA BAC AB DEABC BAC

alternate`

+ + +

+ + +

+ +

<

===

is isoscelesABC` ∆


1.

AB is a diameter of a circle with centre O, and .AC BC= Prove(a) OC is perpendicular to AB.(b) °.ACB 90+ =

2.

A wheel has 9 spokes, equidistant from one another. Find

the angle subtended at the (a) centre between each spoke.

the length of the arc between (b) each spoke in terms of the radius r.

3.

Prove that if two chords subtend equal angles at the centre of a circle, then the chords are equal.

4. given two equal circles with centres O and P, and chords

,AB CD= prove that the chords cut off equal angles at the centres of the circles.

5. .BC DC= Prove .BAC DAC+ +=

1.4 exercisesGenerally, in these exercises O is the centre of the circle.


6. Prove 180 ,°x y+ = given any quadrilateral OABC where O is the centre of the circle, A, B and C are concyclic points, and

,AOC x2+ = ABC y+ = as shown.

7. AB BC= and .BC AD<

Prove (a) AC bisects .BAD+

show (b) BAC+ and ADC+ are complementary.

8. OABC is a quadrilateral with O the centre of the circle and

.AB BC= Prove that BO is perpendicular to AC.

9. show ( ) .XOZ YXO YZO2+ + += +

10. Prove 180 .°BCD BAD+ += −

11. O is the centre of the circle and MP is perpendicular to NQ. Prove(a) QMR RQP+ +=(b) .MN MQ=

12. given ,DAC CDE+ += prove AE bisects .DAB+

Complementary angles add up to 90˚.


13. .EF FG= Prove EFGH is a kite.

14. P is the centre of the larger circle. Prove BC is a diameter of the smaller circle.

15. AB is a diameter of the circle with centre O. Prove .BDC ACO+ +=

16. Prove .AB DE=

17. given ,AB CD= prove .OFE OEF+ +=

18. if ,AB CD= prove AE DE= and .CE BE=

19. given ,BD CD= show(a) ABC∆ is similar to ODB∆(b) .AB OD<

20. Prove .CD FG=


21. (a) Prove OP is perpendicular to AB (O and P are the centres of the circles).

(b) if 18 ,AB cm= 14AP cm= and 26 ,OP cm= find the radius OA to

the nearest cm.

22. (a) Prove CF is a diameter of the small circle and GD is a diameter of the larger circle, given CD is the perpendicular bisector of AB.

(b) if 15AB cm= and 2 ,EF cm= find the length of CE.

23. given ,AB DC< and AD BC< prove ABCD must be a rectangle.

24. BEDF is a cyclic quadrilateral with °.FDE 90+ = Prove AC is a diameter of the larger circle.

25. AC bisects DAB+ and .DCB+ Prove AC is a diameter of the circle.

26. Prove °x y z 180+ + = (O is the centre of the circle).


27. Prove 2 .θADC+ =

28. Prove ADB∆ is similar to .CED∆

29. given ,ED CD= prove .AB EC<

30. Prove F, E and D are collinear, given .AF CD<

31. (a) Prove OAB∆ and OCB∆ are congruent triangles, given O is the centre of the circle with tangents AB and BC.

(b) show .AOB COB+ +=

32. Prove .ACD CED+ +=

33. Prove OABC is a cyclic quadrilateral. AB and CB are tangents to the circle, which has centre O.

O

A

B

C


34. D is the point of contact of two circles. Prove ,AB CB= where AB and CB are tangents to the circles.

A B

C

D

35. Prove .AB CD< AE and BE are tangents to both circles.

36. D is the midpoint of AB. show ,AO BO= where O is the centre of

the circle, and AB is a tangent to the circle.

37. Prove(a) AC BD=(b) .AB DC<

AC and BD are tangents to both circles.

A

D

C

E

B

38. BD bisects ABC+ and .AB AD= Prove ABCD is a rhombus. BC

and DC are tangents.

B

CD

A

39. given two concentric circles with centre O

prove (a) OAB∆ and OCD∆ are similar triangles

find (b) CD if the radii of the circles are 5 cm and 8 cm and

.AB 12 cm=

40. O and P are the centres of the circles below and AC is a tangent to both circles at B. Prove O, P and B are collinear.


41. Prove ,ADC BAC2+ += given that BC bisects .ECA+

42. show that ABC∆ and CDE∆ are similar. O is the centre of the larger circle.

43. Prove .BOA EAB2+ +=

44. CF is perpendicular to AB. Prove BC bisects .FCE+ O is the centre of the circle.

45. if ,DE BA< prove EDC∆ is isosceles.

46. .DE BD4= Prove AB BD5= where AB and AC are tangents.

47. given ,BD CE< show BDE∆ is similar to .BEC∆


48. Prove that the figure below is impossible.

49. Prove(a) EDF BAE+ +=(b) ADB∆ is similar to EDC∆(c) .AB FG<

50. if Y is the midpoint of AB, prove(a) AC BC=(b) AZY∆ is congruent to BXY∆(c) XYZ∆ is congruent to .AZY∆

51. given ,AE OA=prove (a) OC BD<

show (b) EF AE3=if (c) OB 8 cm= and BD 7= cm,

find the length of OC (O is the centre of the larger circle and EF is a tangent to the circle).

C

AE

F

OB

D

52. .DC FG< Prove A, B, C and D are concyclic points.


Problem

What is the wrong assumption made in this proof for angles in the alternate segment?

to prove: ABC CAD+ +=

Proof

90 ( )

90 ( )

180 (90 90 )

°

°

° ° °

CAD x

BAC x

BCA

ABC x

x

CAD ABC

Let

Then tangent radius

in semicircle

`

`

=

+

+

+ +

+

+ +

== −== − + −==


1. triangle ABC is isosceles, with .AB AC=D is the midpoint of AB and E is the midpoint of AC.

Prove that (a) BEC∆ is congruent to BDC∆ .

Prove (b) BE DC= .

2. in the quadrilateral ABCDevaluate (a) x and yprove that (b) AB and DC are parallel.

3. if the diagonals of a rhombus are x and y, show that the length of its side is

.x y

2

2 2+

4. if ( , ), ( , )A B4 1 7 5= − = − and ( , ),C 1 3= prove that triangle ABC is isosceles.

5. the surface area of a closed cylinder is 100 m2. Write the height h of the cylinder in terms of its radius r.

6. ABCD is a parallelogram with ,°C AE ED45 perpendicular to+ = and

CD DE= .

show that (a) ADE∆ is isosceles.if (b) AE y= , show that the area of ABCE

is .y

2

3 2

7. in the circle with centre O, CAO x BAO yand+ += = .

show that 90 ( )°OCB x y+ = − + .

8. in the figure AEF

prove (a) BACB

DECD=

find the length of (b) AE.

Test Yourself 1


9. given ( , ), ( , ),A B1 3 2 4= − = − − ( , )C 5 4= − ( , ),D 6 3and = prove ABCD is a

parallelogram.

10. AC is a tangent to the circle in which BD ED= .

Prove BD EBCbisects+ .

11. A parallelogram has sides 5 cm and 12 cm, with diagonal 13 cm.

show that the parallelogram is a rectangle.

12. Prove that PQR WXYand∆ ∆ are similar.

13. Prove that if two chords subtend equal angles at the centre of a circle, the chords are equal.

14. in quadrilateral ABCE, AD ED DC= =90°ACBand+ = . Also, AC BADbisects+ .

Prove ABCE is a parallelogram.

15. if ( , ), ( , )A B1 5 4 2= = and ( , )C 2 3= − , find the coordinates of D such that ABCD is a parallelogram.

16. (a) Prove that ABC∆ is similar to .CDE∆ (b) evaluate x and y to 1 decimal place.

17. (a) Find the equation of AB if ( , )A 2 3= − − and ( , )B 4 5= .

(b) Find the perpendicular distance from ( , )C 1 3− to line AB.

(c) Find the area of .ABC∆

18. ABCD is a kite.

Prove (a) ABC ADCand∆ ∆ are congruent.Prove (b) ABE ADEand∆ ∆ are congruent.Prove (c) AC is the perpendicular

bisector of BD.


19. ( , ), ( , )A B1 2 3 3 and ( , )C 5 1− are points on a number plane.

show that (a) AB is perpendicular to BC.Find the coordinates of (b) D such that

ABCD is a rectangle.Find the point where the diagonals of (c)

the rectangle intersect.calculate the length of the diagonal.(d)

20. the surface area of a box is 500 cm2. its length is twice its breadth x.

show that the height (a) h of the box is

given by hx

x3

250 2 2

= − .

show that the volume of the box is(b)

.V x x3

500 4 3

= −

Challenge Exercise 1

1. in the figure, BD is the perpendicular bisector of AC. Prove that ABC∆ is isosceles.

2. given E and D are midpoints of AC and AB respectively, prove that(a) DE is parallel to BC

(b) .DE BC21=

3. Prove that the diagonals in a rhombus bisect the angles they make with the sides.

4. Paper comes in different sizes, called A0, A1, A2, A3, A4 and so on. the largest size is A0, which has an area of one square metre. if the ratio of its length to breadth is :2 1, find the dimensions of its sides in millimetres, to the nearest millimetre.

5. the volume of a prism with a square base of side x is 1000 cm3. Find its surface area in terms of x.

6. Prove that in any regular n-sided polygon

the size of each angle is °.n180 360−b l

7. Line XY meets ABC∆ so that .BAC BCY+ += if a circle can be drawn

through A, B and C, show that XY is a tangent to the circle.


8. A plastic frame for a pair of glasses is designed as below. Find the length of plastic needed for the frame, to the nearest centimetre.

9. A circle with centre O has .CO BA< Prove(a) OCB CAD+ +=(b) .°CBA CAO90+ += +

10. A parallelogram ABCD has AB produced to E and diagonal AC produced to F so that .EF BC< Prove that AEF∆ is similar to .ADC∆

11. ABCD is a rhombus with ( , ), ( , ),A B a b0 0( , )C a2 0 and ( , ).D a b− show

the diagonals bisect each other at (a) right angles

all sides are equal (b) Ac bisects (c) .BCD+

12. in the parallelogram ABCD, AC is perpendicular to BD. Prove that .AB AD=

13. triangle ABC has P, Q and R as midpoints of the sides, as shown in the diagram below. Prove that .PQR CPR/∆ ∆

14. A tangent is drawn from C to meet the circle at B. A secant is drawn from C to intersect the circle at A and D. O is the centre of the circle. Prove(a) OBD DCB+ +=(b) .AOD DCB2+ +=


15. two circles have C as a point of contact with common tangent AB. Prove

.DCE FCG+ +=

16. in the circle below, D is the midpoint of AC and O is the centre of the circle. Prove that .AOD CBA+ +=

17. (a) three equal circles with radius r are stacked as shown. Find the height of the stack.

if 6 such circles are stacked in the (b) same way, what will their height be?

Find the height of a stack of 21 such (c) circles.

18. tangents BE and FD are common to the circles with centres A and C. Prove that

90 .°FBD+ =

19. A cart is made with wheels of radii R and r. the distance between the centres of the wheels is .R r3 + Find the length, AB, of the top needed to be put on the cart, if AB is a tangent to both circles.


20. A pair of earrings is made with a wire surround holding a circular stone, as shown. Find the total length of wire needed for the earrings.

21. the sides of a quadrilateral ABCD have midpoints P, Q, R and S, as shown below.

show that (a) DPS∆ is similar to .DAC∆show (b) .PS QR<

show that (c) PQRS is a parallelogram.

chapter 1 - geometry 2

Documents

corresponding angles

pairs of angles

cointerior angles

included angles

equal sas

equal rhs

equal aas

pairs of corresponding