1 eso - unit 10 - lines and angles. geometric figures

Unit 10 April

1. LINES.

1.1. PERPENDICULAR BISECTOR.

In Geometry, Bisection is the division of something into two equal parts, usually

by a Line, which is then called a Perpendicular Bisector or Segment Bisector (a line

that passes through the Midpoint of a given segment).

How to construct a Perpendicular Bisector in a Segment 𝑨𝑨𝑨𝑨��?

• Step 1: Stretch your compasses until it is more than half the length of 𝑨𝑨𝑨𝑨��. Put the

sharp end at 𝑨𝑨 and mark an arc above and another arc below line segment 𝑨𝑨𝑨𝑨��.

• Step 2: Without changing the width of the compasses, put the sharp end at 𝑨𝑨 and

mark arcs above and below the line segment 𝑨𝑨𝑨𝑨�� that will intersect with the arcs

drawn in step 1.

Axel Cotón Gutiérrez Mathematics 1º ESO 10.1

Unit 10 April

• Step 3: Join the two points where the arcs intersect with a straight line. This line is

the perpendicular bisector of 𝑨𝑨𝑨𝑨��. P is the midpoint of 𝑨𝑨𝑨𝑨��.

𝑨𝑨𝑨𝑨�� = 𝑨𝑨𝑨𝑨��

𝑨𝑨𝑨𝑨�� = 𝑨𝑨𝑨𝑨��

Q


Unit 10 April

1.2. ANGLE BISECTOR.

The Angle Bisector is a line that passes through the apex of an angle, that

divides it into two equal angles.

How to construct an Angle Bisector for the following angle?

• Step 1: Put the sharp end of your compasses at point 𝑨𝑨 and make one arc on the

line 𝑨𝑨𝑩𝑩�� (point 𝑺𝑺) and another arc on line 𝑨𝑨𝑨𝑨�� (point 𝑻𝑻).

• Step 2: Put the sharp end of the compasses at 𝑺𝑺 and make an arc within the

lines 𝑨𝑨𝑨𝑨�� and 𝑨𝑨𝑩𝑩��. Do the same at T and make sure that the second arc intersects

the first arc.


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• Step 3: Draw a line from point 𝑨𝑨 to the points of intersection of the 2 arcs. This

line bisects 𝑨𝑨𝑨𝑨𝑩𝑩� .

MATH VOCABULARY: Bisection, Perpendicular Bisector, Segment Bisector, Arc,

Midpoint, Segment, Line, Apex.


Unit 10 April

2. ANGLES.

2.1. PARALLEL ANGLES.

Two Parallel Angles are equal or Supplementary Angles.

Two angles are Supplementary when their addition is 𝟏𝟏𝟏𝟏𝟏𝟏°.

When Parallel Lines get crossed by another line (which is called a Transversal),

you can see that many angles are the same.


Unit 10 April

The pairs of angles on opposite sides of the transversal but outside the two

lines are called Alternate Exterior Angles.

The angles in matching corners are called Corresponding Angles.

The pairs of angles on opposite sides of the transversal but inside the two lines

are called Alternate Interior Angles.


Unit 10 April

The pairs of angles on one side of the transversal but inside the two lines are

called Consecutive Interior Angles.

Vertically Opposite Angles are the angles opposite each other when two lines

cross.

Alternate Exterior Angles, Corresponding Angles, Alternate Interior Angles and

Vertically Opposite Angles are equal.

MATH VOCABULARY: Parallel Angles, Supplementary Angles, Transversal, Alternate

Exterior Angles, Corresponding Angles, Alternate Interior Angles, Consecutive Interior

Angles, Vertically Opposite Angles.


Unit 10 April

2.2. MEASUREMENT OF ANGLES.

2.2.1. SEXAGESIMAL DEGREE.

Remember that a Right Angle is 𝟗𝟗𝟏𝟏°.

A Straight Angle is 𝟏𝟏𝟏𝟏𝟏𝟏 𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅.

A Full Angle is 𝟑𝟑𝟑𝟑𝟏𝟏 𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅.

There are 𝟑𝟑𝟑𝟑𝟏𝟏 𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅 in one Full Rotation (one complete circle around). The

most usual unit of measurement for angles is the Sexagesimal Degree, which consists


Unit 10 April

in 𝟏𝟏/𝟑𝟑𝟑𝟑𝟏𝟏 of a full angle. The measurement of an angle in Sexagesimal Degrees is

denoted by the symbol °. To get an idea, one degree corresponds to the following

aperture:

What happens when we have an angle of less than 𝟏𝟏°?

To be able to speak about angles that measure less than 𝟏𝟏°, we use

submultiples of a degree, so we avoid working with expressions like the following:

• This angle measures half a degree.

• This angle measures 0.56 degrees

Thus, the Sexagesimal Degree has submultiples: these are the Minute and the

Second. The Minute is designated as ‘ and the second as ‘’.


Unit 10 April

The measurement of an angle in degrees, minutes and seconds would be, for

example, 𝟓𝟓𝟓𝟓° 𝟏𝟏𝟓𝟓′ 𝟑𝟑𝟓𝟓′′. It would be read as: an angle of 𝟓𝟓𝟓𝟓 degrees, 𝟏𝟏𝟓𝟓 minutes and

𝟑𝟑𝟓𝟓 seconds.

Let's see the exact value of minutes and seconds:

• One Minute is the result of taking a degree and dividing 𝟑𝟑𝟏𝟏 it into equal parts. This

is, mathematically expressed:

𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝒅𝒅 =𝟏𝟏°𝟑𝟑𝟏𝟏

⇒ 𝟑𝟑𝟏𝟏 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝒅𝒅𝒅𝒅 = 𝟏𝟏°

• A Second is the result of taking a minute and dividing it in 𝟑𝟑𝟏𝟏 equal parts. This is,

mathematically expressed:

𝟏𝟏 𝒅𝒅𝒅𝒅𝒔𝒔𝒔𝒔𝟏𝟏𝒅𝒅 =𝟏𝟏′𝟑𝟑𝟏𝟏

⇒ 𝟑𝟑𝟏𝟏 𝒅𝒅𝒅𝒅𝒔𝒔𝒔𝒔𝟏𝟏𝒅𝒅𝒅𝒅 = 𝟏𝟏′

2.2.2. USING A PROTRACTOR.

To find the number of degrees in an angle, we use a Protractor.

A straight angle is divided into 𝟏𝟏𝟏𝟏𝟏𝟏 equal subdivisions on a Protractor, marked

from 𝟏𝟏 to 𝟏𝟏𝟏𝟏𝟏𝟏. You find two scales marked 𝟏𝟏 to 𝟏𝟏𝟏𝟏𝟏𝟏, one in clockwise direction and

the other in anti-clockwise direction. Each subdivision stands for 𝟏𝟏°.

Steps to measure:

• Place the center of the protractor on the vertex of the angle.

• Base line should fall along any of the sides.


Unit 10 April

• The scale (clock wise or anti-clock wise direction), which begins with zero on the

side, is chosen.

• Read the mark on the scale where the other arm crosses it.

2.2.3. COMPLEX AND NON-COMPLEX FORM.

To change degrees into minutes and seconds we will always work by means of

Conversion Factors. To convert from Complex Form to Non-Complex Form:

𝑴𝑴𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝒅𝒅𝒅𝒅 = 𝑫𝑫𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅 ∙ 𝟑𝟑𝟏𝟏

𝑺𝑺𝒅𝒅𝒔𝒔𝒔𝒔𝟏𝟏𝒅𝒅𝒅𝒅 = 𝑴𝑴𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝒅𝒅𝒅𝒅 ∙ 𝟑𝟑𝟏𝟏

𝑺𝑺𝒅𝒅𝒔𝒔𝒔𝒔𝟏𝟏𝒅𝒅𝒅𝒅 = 𝑫𝑫𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅 ∙ 𝟑𝟑,𝟑𝟑𝟏𝟏𝟏𝟏

Write 73° 13′ 48′′ in seconds:

73° = 73 ∙ 3,600′′ = 262,800′′

13′ = 13 ∙ 60′′ = 780´´

73° 13′ 48′′ = 262,800′′ + 780′′ = 263,628′′


Unit 10 April

To do it backwards:

𝑴𝑴𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝒅𝒅𝒅𝒅 =𝑺𝑺𝒅𝒅𝒔𝒔𝒔𝒔𝟏𝟏𝒅𝒅𝒅𝒅

𝟑𝟑𝟏𝟏;𝑹𝑹𝒅𝒅𝟏𝟏𝑹𝑹𝟏𝟏𝟏𝟏𝒅𝒅𝒅𝒅𝒅𝒅 = ′′

𝑫𝑫𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅 =𝑴𝑴𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝒅𝒅𝒅𝒅

𝟑𝟑𝟏𝟏;𝑹𝑹𝒅𝒅𝟏𝟏𝑹𝑹𝟏𝟏𝟏𝟏𝒅𝒅𝒅𝒅𝒅𝒅 = ′

𝑫𝑫𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅 =𝑺𝑺𝒅𝒅𝒔𝒔𝒔𝒔𝟏𝟏𝒅𝒅𝒅𝒅𝟑𝟑,𝟑𝟑𝟏𝟏𝟏𝟏

;𝑹𝑹𝒅𝒅𝟏𝟏𝑹𝑹𝟏𝟏𝟏𝟏𝒅𝒅𝒅𝒅𝒅𝒅 = ′′

Write 263,628′′ in complex form:

263,628′′ ÷ 3,600 = 73°,𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 = 828´´

828´´ ÷ 60 = 13′,𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 = 48′′

Therefore:

263,628′′ = 73° 13′ 48′′

MATH VOCABULARY: Right Angle, Straight Angle, Sexagesimal Degree, Full Angle,

Minute, Second, Protractor.

2.3. ANGLES OPERATIONS.

2.3.1. ADDING ANGLES USING THE SEXAGESIMAL MEASURE.

To Add we need to add separately degrees or hours, minutes and seconds and

then convert the seconds into minutes and the minutes into degrees/hours if we get

more than 𝟑𝟑𝟏𝟏.


Unit 10 April

Add 15° 43′ 30′′ + 25° 50′ 34′′

15° 43′ 30′′+ 25° 50′ 34′′

40° 93′ 64′′+1′ −60′′

40° 94′ 4′′+1° −60′41° 34′ 4′′

2.3.2. SUBTRACTING ANGLES USING THE SEXAGESIMAL MEASURE.

To Subtract we need to subtract separately degrees/hours, minutes and

seconds, if we do not have enough seconds or minutes we convert one degree/hour

into minutes or a minute into seconds.

Subtract 56° 38′11′′ − 32° 43′ 56′′

56° 38′ 11′′− 32° 43′ 56′′

56° 38′ 11′′−1° +𝟓𝟓𝟗𝟗′ +60′′

55° 97′ 71′′− 32° 43′ 56′′

23° 54′ 15′′

We obtain 𝟓𝟓𝟗𝟗′ = +60′ − 1′, 60′ from subtracting 1° from degrees and 1′ by

given +60′′ to seconds.

2.3.3. MULTIPLICATION BY A NATURAL NUMBER.

We Multiply separately degrees/hours, minutes and seconds and then we

convert the seconds into minutes and the minutes into degrees/hours when we get

more than 𝟑𝟑𝟏𝟏 subunits.


Unit 10 April

Multiply (13° 23′ 26′′) · 4:

13° 23′ 26′′𝑥𝑥 452° 92′ 104′′

+1′ −60′′52° 93′ 44′′+1° −60′53° 33′ 44′′

2.3.4. DIVISION BY A NATURAL NUMBER.

We Divide the degrees/hours, and the remainder is converted into minutes

that must be added to the previous quantity that we had, divide the minutes and we

repeat the same that we have before. The remainder is in seconds.

2.4. INTERIOR ANGLES OF POLYGONS.

An Interior Angle is an angle inside a Shape.


Unit 10 April

2.4.1. TRIANGLES.

The Interior Angles of a Triangle add up to 𝟏𝟏𝟏𝟏𝟏𝟏°

60° + 90° + 30° = 180°

70° + 80° + 30° = 180°


Unit 10 April

2.4.2. QUADRILATERALS.

A Quadrilateral has 4 straight sides. The Interior Angles of a Quadrilateral add

up to 𝟑𝟑𝟑𝟑𝟏𝟏°.

90° + 90° + 90° + 90° = 360°

100° + 80° + 90° + 90° = 360°

Because there are 2 triangles in a square: 𝟏𝟏𝟏𝟏𝟏𝟏° + 𝟏𝟏𝟏𝟏𝟏𝟏° = 𝟑𝟑𝟑𝟑𝟏𝟏°


Unit 10 April

2.4.3. PENTAGON.

A Pentagon has 5 sides, and can be made from three triangles. Its interior

angles add up to 𝟑𝟑 × 𝟏𝟏𝟏𝟏𝟏𝟏° = 𝟓𝟓𝟓𝟓𝟏𝟏°. And when it is regular (all angles the same),

then each angle is 𝟓𝟓𝟓𝟓𝟏𝟏° / 𝟓𝟓 = 𝟏𝟏𝟏𝟏𝟏𝟏°.

The Interior Angles of a Pentagon add up to 𝟓𝟓𝟓𝟓𝟏𝟏°.

2.4.4. THE GENERAL RULE.

Each time we add a side (triangle to quadrilateral, quadrilateral to pentagon,

etc), we add another 𝟏𝟏𝟏𝟏𝟏𝟏° to the total:


Unit 10 April

𝑺𝑺𝟏𝟏𝟏𝟏 𝒔𝒔𝒐𝒐 𝑰𝑰𝟏𝟏𝟏𝟏𝒅𝒅𝒅𝒅𝟏𝟏𝒔𝒔𝒅𝒅 𝑨𝑨𝟏𝟏𝒅𝒅𝑨𝑨𝒅𝒅𝒅𝒅 = (𝟏𝟏 − 𝟐𝟐) × 𝟏𝟏𝟏𝟏𝟏𝟏°

𝑬𝑬𝑹𝑹𝒔𝒔𝑬𝑬 𝑨𝑨𝟏𝟏𝒅𝒅𝑨𝑨𝒅𝒅 (𝒔𝒔𝒐𝒐 𝑹𝑹 𝑹𝑹𝒅𝒅𝒅𝒅𝟏𝟏𝑨𝑨𝑹𝑹𝒅𝒅 𝑨𝑨𝒔𝒔𝑨𝑨𝑷𝑷𝒅𝒅𝒔𝒔𝟏𝟏) = (𝟏𝟏 − 𝟐𝟐) × 𝟏𝟏𝟏𝟏𝟏𝟏°

𝟏𝟏

2.5. CIRCUMFERENCE ANGLES.

2.5.1. CENTRAL ANGLE.

Central Angle is the angle subtended at the center of a circle by two given

points on the circle.

2.5.2. INSCRIBED ANGLE.

Inscribed Angle is an angle made from points sitting on the circle's

circumference.

𝑨𝑨 and 𝑩𝑩 are "End Points"

𝑨𝑨 is the "Apex Point"


Unit 10 April

2.5.3. INSCRIBED ANGLE THEOREMS.

An Inscribed Angle 𝑹𝑹° is half of the Central Angle 𝟐𝟐𝑹𝑹. Called The Angle at the

Center Theorem.

And (keeping the endpoints fixed) the angle 𝑹𝑹° is always the same, no matter

where it is on the circumference the angle 𝑹𝑹° is the same. Called The Angles

Subtended by Same Arc Theorem.


Unit 10 April

2.5.4. ANGLE IN A SEMICIRCLE.

An Angle Inscribed in a Semicircle is always a Right Angle. The end points are

either end of a circle's diameter, the apex point can be anywhere on the

circumference.

The inscribed angle 𝟗𝟗𝟏𝟏° is half of the central angle 𝟏𝟏𝟏𝟏𝟏𝟏°.

MATH VOCABULARY: Polygon, Interior Angle, Shape, Triangle, Quadrilateral, Square,

Side, Pentagon, Hexagon, Heptagon, Octagon, Nonagon, Decagon, Circumference,

Central Angle, Circle, Arc, Inscribed Angle, End Points, Apex Point, Theorem, Semicircle.


Unit 10 April

3. LINES OF SYMMETRY OF PLANE SHAPES.

3.1. LINE OF SYMMETRY.

Another name for Reflection Symmetry. One half is the reflection of the other

half. The "Line of Symmetry" is the imaginary line where you could fold the image and

have both halves match exactly.

You can find if a shape has a Line of Symmetry by folding it. When the folded

part sits perfectly on top (all edges matching), then the fold line is a Line of Symmetry.

Here I have folded a rectangle one way, and it didn't work.

But when I try it this way, it does work (the folded part sits perfectly on top, all

edges matching):


Unit 10 April

3.2. TRIANGLES.

A Triangle can have 𝟑𝟑, or 𝟏𝟏 or 𝟏𝟏𝒔𝒔 𝑨𝑨𝟏𝟏𝟏𝟏𝒅𝒅𝒅𝒅 of symmetry:

3.3. QUADRILATERALS.

Different types of Quadrilaterals (a 4-sided plane shape):


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3.4. REGULAR POLYGONS.

A Regular Polygon has all sides equal, and all angles equal:

And the pattern continues:

• A regular polygon of 9 sides has 9 Lines of Symmetry

• A regular polygon of 10 sides has 10 Lines of Symmetry

• ...

• A regular polygon of "n" sides has "n" Lines of Symmetry


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3.5. CIRCLE.

A line (drawn at any angle) that goes through its center is a Line of Symmetry.

So a Circle has infinite Lines of Symmetry.

3.6. PLANE SHAPES.

If a Plane Shape has 𝟏𝟏 Lines of Symmetry, all of then cut in one point, and

every two near lines form an angle:

𝜶𝜶 =𝟏𝟏𝟏𝟏𝟏𝟏°𝟏𝟏

𝜶𝜶 =𝟏𝟏𝟏𝟏𝟏𝟏°𝟓𝟓

= 𝟓𝟓𝟓𝟓°

MATH VOCABULARY: Symmetry, Plane Shapes, Line of Symmetry, Reflection

Symmetry, Rhombus.


Unit 10 April

4. PLANE FIGURES.

4.1. TRIANGLES.

4.1.1. EQUILATERAL, ISOSCELES AND SCALENE.

There are three special names given to triangles that tell how many sides (or

angles) are equal. There can be 3, 2 or no equal sides/angles.

4.1.2. ACUTE, OBTUSE AND RIGHT ANGLES.

Triangles can also have names that tell you what type of angle is inside:


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4.1.3. RELATIONSHIP OF SIDES TO INTERIOR ANGLES IN A TRIANGLE.

In a Triangle:

• The shortest side is always opposite the smallest interior angle.

• The longest side is always opposite the largest interior angle.

Recall that in a Scalene Triangle, all the sides have different lengths and all the

interior angles have different measures. In such a triangle, the shortest side is always

opposite the smallest angle. (These are shown in bold color above)


Unit 10 April

Similarly, the longest side is opposite the largest angle. If the smallest side is

opposite the smallest angle, and the longest is opposite the largest angle, then it

follows that since a triangle only has three sides, the midsize side is opposite the

midsize angle.

An Equilateral Triangle has all sides equal in length and all interior angles equal.

Therefore there is no "largest" or "smallest" in this case.

Isosceles Triangles have two sides the same length and two equal interior

angles. Therefore there can be two sides and angles that can be the "largest" or the

"smallest".

4.1.4. CONSTRUCT A TRIANGLE GIVEN THE LENGTH OF ITS THREE SIDES.

We can use a pair of compasses and a ruler to construct a triangle when the

lengths of its sides are given.

Construct a triangle ∆𝐏𝐏𝐏𝐏𝐏𝐏 with 𝐏𝐏𝐏𝐏�� = 𝐱𝐱 𝐜𝐜𝐜𝐜, 𝐏𝐏𝐏𝐏�� = 𝐲𝐲 𝐜𝐜𝐜𝐜 and 𝐏𝐏𝐏𝐏�� = 𝐳𝐳 𝐜𝐜𝐜𝐜.

• Step 1: Draw a line segment 𝐏𝐏𝐏𝐏�� with lengths 𝐱𝐱 𝐜𝐜𝐜𝐜 and mark the points 𝑨𝑨 and 𝑨𝑨.

• Step 2: To draw 𝐏𝐏𝐏𝐏��, first stretch the 2 arms of the compasses 𝐲𝐲 𝐜𝐜𝐜𝐜 apart, place

the sharp point at point 𝑨𝑨 and mark an arc with the pencil end.

x cm


Unit 10 April

• Step 3: To draw 𝐏𝐏𝐏𝐏��, adjust the compasses to 𝐳𝐳 𝐜𝐜𝐜𝐜, place the sharp point at

point 𝑨𝑨 and mark an arc with the pencil end. You need to draw the arc so that it

will intersect with the arc drawn in step 2. Label the point of intersection as

point 𝑹𝑹.

• Step 4: Draw straight lines from 𝑨𝑨 to 𝑹𝑹 and from 𝑨𝑨 to 𝑹𝑹.

x cm

x cm

x cm


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4.1.5. MEDIAN. CENTROID.

A Median of a Triangle is a line segment from a vertex (corner point) to the

midpoint of the opposite side.

A triangle has three medians, and they all cross over at a special point called

the "Centroid".

4.1.6. ALTITUDE. ORTHOCENTER.

Altitude is another word for height. In a triangle is the line at right angles to a

side that goes through the opposite corner.


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Where all three lines intersect is the "Orthocenter":

Note that sometimes the edges of the triangle have to be extended outside the

triangle to draw the Altitudes. Then the Orthocenter is also outside the triangle.

4.1.7. CIRCUMCENTER.

Using the Perpendicular Bisector of sides:

Where all three perpendicular bisectors intersect is the center of a triangle's

"Circumcircle", called the "Circumcenter":


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4.1.8. INCENTER.

If we use the Angle Bisector of the angles of the triangle:

Where all three angle bisectors intersect is the center of a triangle's "Incircle",

called the "Incenter"


Unit 10 April

4.1.9. PYTHAGORAS' THEOREM.

When a triangle has a Right Angle (90°), and squares are made on each of the

three sides, then the biggest square has the exact same area as the other two squares

put together!

It is called "Pythagoras' Theorem" and can be written in one short equation

Note: 𝒔𝒔 is the longest side of the triangle, 𝑹𝑹 and 𝒃𝒃 are the other two sides.


Unit 10 April

The longest side of the triangle is called the "Hypotenuse", so the formal

definition is:

In a right angled triangle: the square of the hypotenuse is equal to the sum of the

squares of the other two sides.

52 = 32 + 42

25 = 9 + 16 = 25


Unit 10 April

There are many geometric problems where we have to use the Pythagoras'

Theorem:

We have to study the shape to find out the right angles and the right triangles,

and apply the Pythagoras' Theorem:


Unit 10 April

Rhombi

Trapezoids

Regular Polygons

Circles


Unit 10 April

4.1.10. AREA AND PERIMETER.

The Perimeter is the distance around the edge of the triangle: just add up the

three sides:

208 + 145 + 203 = 556

The Area is half of the base times height. "𝒃𝒃" is the distance along the Base. "𝑬𝑬"

is the height (measured at right angles to the base).

𝑨𝑨 =𝒃𝒃 ∙ 𝑬𝑬𝟐𝟐


Unit 10 April

Many times we will have to use the Pythagoras' Theorem to find the height of

the triangle.

ℎ2 + 52 = 102

ℎ2 = 102 − 52 = 100 − 25 = 75

ℎ = √75 ≈ 8.7 𝑐𝑐𝑅𝑅

𝐴𝐴 =10 ∙ 8,7

2= 43.5 𝑐𝑐𝑅𝑅2

MATH VOCABULARY: Plane Figures, Equilateral, Isosceles, Scalene, Acute, Obtuse,

Median, Centroid, Altitude, Orthocenter, Circumcenter, Incenter, Pythagoras' Theorem,

Hypotenuse, Rhombus, Trapezoid, Area, Perimeter, Base.


Unit 10 April

4.2. QUADRILATERALS.

Quadrilateral just means "four sides" (quad means four, lateral means side). A

Quadrilateral has four-sides, it is 2-dimensional (a flat shape), closed (the lines join

up), and has straight sides.


Unit 10 April

4.2.1. SQUARE.

A Square is a flat shape with 4 equal sides and every angle is a right angle (90°).

Properties:

• All sides are equal in length.

• Each internal angle is 90°.

• Opposite sides are parallel (so it is a Parallelogram).

The Area is the side length squared:

𝑨𝑨𝒅𝒅𝒅𝒅𝑹𝑹 = 𝑹𝑹 ∙ 𝑹𝑹 = 𝑹𝑹𝟐𝟐

The Area is also half of the diagonal squared:

𝑨𝑨𝒅𝒅𝒅𝒅𝑹𝑹 =𝒅𝒅𝟐𝟐

𝟐𝟐

The Perimeter is 4 times the side length:


Unit 10 April

𝑨𝑨 = 𝑹𝑹 + 𝑹𝑹 + 𝑹𝑹 + 𝑹𝑹 = 𝟓𝟓 ∙ 𝑹𝑹

A square has two Diagonals; they are equal in length and intersect in the

middle. The Diagonal is the side length times the square root of 2:

𝒅𝒅 = 𝑹𝑹 ∙ √𝟐𝟐

4.2.2. RECTANGLE.

A Rectangle is a four-sided flat shape where every angle is a right angle (90°).

Properties:

• Each internal angle is 90°.

• Opposite sides are parallel (so it is a Parallelogram).

The Area is the width times the height:

𝑨𝑨 = 𝒘𝒘 ∙ 𝑬𝑬


Unit 10 April

The Perimeter is the distance around the edges. The Perimeter is 2 times the

(width + height):

𝑨𝑨 = 𝟐𝟐𝒘𝒘 + 𝟐𝟐𝑬𝑬 = 𝟐𝟐(𝒘𝒘 + 𝑬𝑬)

A rectangle has two Diagonals, they are equal in length and intersect in the

middle.

The Diagonal is the square root of (width squared + height squared):

𝒅𝒅 = �𝒘𝒘𝟐𝟐 + 𝑬𝑬𝟐𝟐

4.2.3. RHOMBUS.

A Rhombus is a flat shape with 4 equal straight sides.


Unit 10 April

Properties:

• All sides have equal length.

• Opposite sides are parallel, and opposite angles are equal (it is a Parallelogram).

• The Altitude is the distance at right angles to two sides.

• And the Diagonals "𝒑𝒑" and "𝒒𝒒" of a rhombus bisect each other at right angles.

• The Square is a Rhombus.

The Area can be calculated by:

• The altitude times the side length:

𝑨𝑨 = 𝑹𝑹𝑨𝑨𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝒅𝒅𝒅𝒅 ∙ 𝒅𝒅

• Multiplying the lengths of the diagonals and then dividing by 2:

𝑨𝑨 = 𝒑𝒑 ∙ 𝒒𝒒𝟐𝟐

The Perimeter is 4 times "s" (the side length) because all sides are equal in

length:

𝑨𝑨 = 𝟓𝟓 ∙ 𝒅𝒅


Unit 10 April

It is more common to call this shape a Rhombus, but some people call it a

Rhomb or even a Diamond. The plural is Rhombi or Rhombuses, and, rarely, Rhombbi

or Rhombbuses (with a double b).

4.2.4. PARALLELOGRAM.

A Parallelogram is a flat shape with opposite sides parallel and equal in length.

Properties:

• Opposite sides are parallel.

• Opposite sides are equal in length.

• Opposite angles are equal (angles "𝑹𝑹" are the same, and angles "𝒃𝒃" are the same).

• Angles "𝑹𝑹" and "𝒃𝒃" add up to 𝟏𝟏𝟏𝟏𝟏𝟏°, so they are supplementary angles.

• A Parallelogram where all angles are right angles is a rectangle.

The Area is the base times the height:

𝑨𝑨 = 𝒃𝒃 ∙ 𝑬𝑬

The Perimeter is 2 times the (base + side length):


Unit 10 April

𝑨𝑨 = 𝟐𝟐𝒃𝒃 + 𝟐𝟐𝒅𝒅 = 𝟐𝟐(𝒃𝒃 + 𝒅𝒅)

The Diagonals of a parallelogram bisect each other. In other words the

diagonals intersect each other at the half-way point.

4.2.5. TRAPEZOID.

A Trapezoid is a 4-sided flat shape with straight sides that has a pair of opposite

sides parallel (marked with arrows below):

Properties:

• Has a pair of parallel sides.

• Is an Isosceles Trapezoid when both angles coming from a parallel side are equal,

and the sides that aren't parallel are equal in length.

• Is called a "Trapezium" in the UK.


Unit 10 April

The Area is the average of the two base lengths times the altitude:

𝑨𝑨 =𝑹𝑹 ∙ 𝒃𝒃𝟐𝟐

∙ 𝑬𝑬

The Perimeter is the sum of all side lengths:

𝑨𝑨 = 𝑹𝑹 + 𝒃𝒃 + 𝒔𝒔 + 𝒅𝒅

The Median (also called a midline or Midsegment) is a line segment half-way

between the two bases.

𝟏𝟏 =𝑹𝑹 + 𝒃𝒃𝟐𝟐

You can calculate the area when you know the median, it is just the median

times the height:

𝑨𝑨 = 𝟏𝟏 ∙ 𝑬𝑬


Unit 10 April

A Trapezium (UK: Trapezoid) is a quadrilateral with NO parallel sides. The US

and UK have their definitions swapped over.

4.2.6. KITE.

A Kite is a flat shape with straight sides. It has 2 pairs of equal adjacent sides.

Properties:

• Two pairs of sides.

• Each pair is made up of adjacent sides (they meet) that are equal in length.

• The angles are equal where the two pairs meet.

• Diagonals (dashed lines) cross at right angles, and one of the diagonals bisects

(cuts equally in half) the other.

• When all sides have equal length the Kite will also be a Rhombus.

• When all the angles are also 𝟗𝟗𝟏𝟏° the Kite will be a Square.

To find the Area of a Kite, Multiply the lengths of the diagonals and then divide

by 2 to find the Area:

𝑨𝑨 =𝒑𝒑 ∙ 𝒒𝒒𝟐𝟐


Unit 10 April

The Perimeter is 2 times (side length a + side length b):

𝑨𝑨 = 𝟐𝟐(𝑹𝑹 + 𝒃𝒃)

A concave Kite is called a Dart.

MATH VOCABULARY: Diagonal, Rectangle, Rhomb, Diamond, Rhombi, Rhombuses,

Rhombbi, Rhombbuses, Parallelogram, Trapezoid, Trapezium, Midsegment, Kite, Dart.

4.3. POLYGONS.

A Polygon is a plane shape with straight sides. Polygons are 2-dimensional

shapes. They are made of straight lines, and the shape is "closed" (all the lines connect

up).


Unit 10 April

A Regular Polygon has all angles equal and all sides equal, otherwise it is

Irregular.

A Convex Polygon has no angles pointing inwards. More precisely, no internal

angle can be more than 𝟏𝟏𝟏𝟏𝟏𝟏°. If any internal angle is greater than 𝟏𝟏𝟏𝟏𝟏𝟏° then the

polygon is Concave. (Think: concave has a "cave" in it).

A Simple Polygon has only one boundary, and it doesn't cross over itself.

A complex polygon intersects itself! Many rules about polygons don't work when it is

Complex.


Unit 10 April


Unit 10 April

4.3.1. CIRCUMCIRCLE, INCIRCLE, RADIUS AND APOTHEM OF REGULAR POLYGONS.

They are just the names of the "Outer" and "Inner" circles (and each Radius)

that can be drawn on a polygon like this:

The "outside" circle is called a Circumcircle, and it connects all vertices (corner

points) of the polygon. The Radius of the circumcircle is also the radius of the polygon.

The "inside" circle is called an Incircle and it just touches each side of the polygon at its

midpoint. The radius of the incircle is the Apothem of the polygon.

4.3.2. AREA AND PERIMETER OF REGULAR POLYGONS.

The Perimeter is 𝟏𝟏 times the Side, where 𝟏𝟏 is the number of sides.

𝑨𝑨 = 𝒅𝒅 ∙ 𝟏𝟏

We can learn a lot about regular polygons by breaking them into triangles like

this:


Unit 10 April

Notice that:

• The "Base" of the triangle is one Side of the polygon.

• The "Height" of the triangle is the "Apothem" of the polygon.

Now, the Area of a Triangle is half of the base times height, so:

𝑨𝑨 =𝒃𝒃 ∙ 𝑬𝑬𝟐𝟐

=𝒅𝒅 ∙ 𝑹𝑹𝒑𝒑𝑬𝑬𝒔𝒔𝟏𝟏𝒅𝒅𝟏𝟏

𝟐𝟐

To get the area of the whole polygon, just add up the areas of all the little

triangles ("𝟏𝟏" of them)

𝑨𝑨𝒅𝒅𝒅𝒅𝑹𝑹 𝒔𝒔𝒐𝒐 𝑨𝑨𝒔𝒔𝑨𝑨𝑷𝑷𝒅𝒅𝒔𝒔𝟏𝟏 =𝒅𝒅 ∙ 𝑹𝑹𝒑𝒑𝑬𝑬𝒔𝒔𝟏𝟏𝒅𝒅𝟏𝟏

𝟐𝟐∙ 𝟏𝟏 =

𝑨𝑨 ∙ 𝑹𝑹𝒑𝒑𝑬𝑬𝒔𝒔𝟏𝟏𝒅𝒅𝟏𝟏𝟐𝟐

MATH VOCABULARY: Convex, Concave, Apothem, Radius.

4.4. CIRCLE.

A Circle is easy to make: Draw a curve that is "Radius" away from a central

point. All points are the same distance from the center.


Unit 10 April

4.4.1. AREA AND CIRCUMFERENCE.

The Radius is the distance from the center outwards. The Diameter goes

straight across the circle, through the center. The Circumference is the distance once

around the circle.

𝑩𝑩𝟏𝟏𝒅𝒅𝒔𝒔𝟏𝟏𝟏𝟏𝒐𝒐𝒅𝒅𝒅𝒅𝒅𝒅𝟏𝟏𝒔𝒔𝒅𝒅 = 𝝅𝝅 ∙ 𝑫𝑫𝟏𝟏𝑹𝑹𝟏𝟏𝒅𝒅𝟏𝟏𝒅𝒅𝒅𝒅

Also note that the Diameter is twice the Radius:

𝑫𝑫𝟏𝟏𝑹𝑹𝟏𝟏𝒅𝒅𝟏𝟏𝒅𝒅𝒅𝒅 = 𝟐𝟐 ∙ 𝑹𝑹𝑹𝑹𝒅𝒅𝟏𝟏𝟏𝟏𝒅𝒅

𝑩𝑩𝟏𝟏𝒅𝒅𝒔𝒔𝟏𝟏𝟏𝟏𝒐𝒐𝒅𝒅𝒅𝒅𝒅𝒅𝟏𝟏𝒔𝒔𝒅𝒅 = 𝟐𝟐 ∙ 𝝅𝝅 ∙ 𝑹𝑹𝑹𝑹𝒅𝒅𝟏𝟏𝟏𝟏𝒅𝒅


Unit 10 April

The Area of a circle is 𝝅𝝅 times the radius squared, which is written:

𝑨𝑨 = 𝝅𝝅 ∙ 𝒅𝒅𝟐𝟐

4.4.2. LINES AND SLICES.

A line that goes from one point to another on the circle's circumference is

called a Chord. If that line passes through the center it is called a Diameter. A line that

"just touches" the circle as it passes by is called a Tangent. And a part of the

circumference is called an Arc.

There are two main "Slices" of a circle. The "pizza" slice is called a Sector. And

the slice made by a chord is called a Segment


Unit 10 April

The Quadrant and Semicircle are two special types of Sector: Quarter of a circle

is called a Quadrant. Half a circle is called a Semicircle.

You can work out the Area of a Sector by comparing its angle to the angle of a

full circle.

𝑨𝑨𝒅𝒅𝒅𝒅𝑹𝑹 𝒔𝒔𝒐𝒐 𝑺𝑺𝒅𝒅𝒔𝒔𝟏𝟏𝒔𝒔𝒅𝒅 = 𝛉𝛉 ∙ 𝛑𝛑 ∙ 𝒅𝒅𝟐𝟐

𝟑𝟑𝟑𝟑𝟏𝟏 (𝒘𝒘𝑬𝑬𝒅𝒅𝟏𝟏 𝜽𝜽 𝟏𝟏𝒅𝒅 𝟏𝟏𝟏𝟏 𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅)

The Area of a Segment is the area of a sector minus the triangular piece (shown

in light blue here).

𝑨𝑨𝒅𝒅𝒅𝒅𝑹𝑹 𝒔𝒔𝒐𝒐 𝑺𝑺𝒅𝒅𝒅𝒅𝟏𝟏𝒅𝒅𝟏𝟏𝟏𝟏 = 𝑨𝑨𝒅𝒅𝒅𝒅𝑹𝑹 𝒔𝒔𝒐𝒐 𝑺𝑺𝒅𝒅𝒔𝒔𝟏𝟏𝒔𝒔𝒅𝒅 − 𝑨𝑨𝒅𝒅𝒅𝒅𝑹𝑹 𝒔𝒔𝒐𝒐 𝑻𝑻𝒅𝒅𝟏𝟏𝑹𝑹𝟏𝟏𝒅𝒅𝑨𝑨𝒅𝒅

The Arc Length (of a Sector or Segment) is:


Unit 10 April

𝑳𝑳 = 𝟐𝟐 ∙ 𝝅𝝅 ∙ 𝒅𝒅 ∙ 𝜽𝜽

𝟑𝟑𝟑𝟑𝟏𝟏 (𝒘𝒘𝑬𝑬𝒅𝒅𝟏𝟏 𝜽𝜽 𝟏𝟏𝒅𝒅 𝟏𝟏𝟏𝟏 𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅)

4.4.3. RELATIVE POSITIONS.

The Circle has different Relative Positions:

• Relative Position of a Straight Line with respect to a circumference:


Unit 10 April

• Relative Position of two circumferences:

MATH VOCABULARY: Convex, Center, Diameter, Slice, Chord, Tangent, Arc, Sector,

Segment, Quadrant, Semicircle.


Unit 10 April

5. SOLID GEOMETRY.

Solid Geometry is the geometry of three-dimensional space, the kind of space

we live in. It is called three-dimensional, or 3D, because there are three dimensions:

Width, Depth and Height.

Let us start with some of the simplest shapes:


Unit 10 April

There are two main types of solids, "Polyhedra", and "Non-Polyhedra": The

Polyhedra must have flat faces. If not they are Non-Polyhedra.

5.1. POLYHEDRONS.

A Polyhedron is a solid with flat faces. So no curved surfaces: cones, spheres

and cylinders are not polyhedrons.

5.1.1. THE PLATONIC SOLIDS.

A Platonic Solid is a 3D shape where:

• Each face is the same Regular Polygon.

• The same number of polygons meet at each vertex (corner).


Unit 10 April

5.1.2. PRISMS.

A Prism is a solid object with:

• Identical ends.

• Flat faces.

• The same cross section all along its length.

A Cross Section is the shape made by cutting straight across an object.


Unit 10 April

The Cross Section of this object is a triangle it has the same cross section all

along its length so it's a Triangular Prism. The ends of a prism are parallel each one is

called a Base.

The side faces of a prism are Parallelograms (4-sided shapes with opposite

sides parallel).

These are all Prisms:


Unit 10 April

All the previous examples are Regular Prisms, because the cross section is

regular (in other words it is a shape with equal edge lengths, and equal angles.). Here is

an example of an Irregular Prism:


Unit 10 April

5.1.3. PYRAMIDS.

A Pyramid is made by connecting a base to an Apex:

There are many types of Pyramids, and they are named after the shape of their

Base.


Unit 10 April

This tells us where the top (apex) of the pyramid is. When the apex is directly

above the center of the base it is a Right Pyramid, otherwise it is an Oblique Pyramid.

This tells us about the shape of the base. When the base is a regular polygon it

is a Regular Pyramid, otherwise it is an Irregular Pyramid.


Unit 10 April

5.2. NON-POLYHEDRA.

5.2.1. SPHERE.

Properties:

• It is perfectly symmetrical.

• All points on the surface are the same distance "r" from the center.

• It has no edges or vertices (corners).

• It has one surface (not a "face" as it isn't flat).

• It is not a polyhedron.

• A Sphere is a Rotated Circle.

5.2.2. CYLINDER.

Properties:

• It has a flat Base and a flat top.


Unit 10 April

• The base is the same as the top.

• From base to top the shape stays the same.

• It has one curved side.

• It is not a polyhedron as it has a curved surface.

• A Cylinder is a Rotated Rectangle.

5.2.3. CONE.

Properties:

• It has a flat base.

• It has one curved side.

• It is not a polyhedron as it has a curved surface.

• The pointy end of a cone is called the Apex.

• The flat part is the Base

• A Cone is a Rotated Triangle.

MATH VOCABULARY: 3D Shapes, Sphere, Torus, Cylinder, Cone, Cube, Cuboid, Pyramid,

Prism, Polyhedra, Non-Polyhedra, Polyhedron, Platonic Solids, Edge, Face, Vertices,

Vertex, Apex.


1 eso - unit 10 - lines and angles. geometric figures

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