geometry capsule

Upload: mousamjha

Post on 23-Feb-2018

236 views

Category:

Documents


0 download

TRANSCRIPT

  • 7/24/2019 Geometry Capsule

    1/13

  • 7/24/2019 Geometry Capsule

    2/13

  • 7/24/2019 Geometry Capsule

    3/13

    Triangles and Their Properties

    A. On the basis of sides, triangles are classified into three categories:

    a) Scalene:Having all sides unequal.

    b) Isosceles:Having any two sides of same length.

    c) Equilateral: Having all the three sides of equal length.

    On the basis of angles, triangles are divided into three categories:

    a) Obtuse angled triangle:Largest angle greater than 900

    b) Acute angled triangle: All angles less than 900

    c) Right Angled Triangle: Largest angle equal to 900

    Properties of a Triangle:

    1.Sum of the all the three angles is 1800.

    2.An exterior angle is equal to the sum of the interior opposite angles.

    3.The sum of any two sides is always greater than the length of the third side.

    4.The difference between any two sides is always less than that of the third side.

    5.The side opposite to the greatest angle is the greatest side and the side opposite to the smallest angle is the

    shortest side

    Points inside or outside a triangle with their properties:

    A. Centroid:The point of intersection of the medians of a triangle

    1.The centroid divides each median from the vertex in the ratio 2 : 1.

    2.Apollonius theorem gives the length of the median.

    AB2+ AC2= 2(AD2+ BD2)

    3.If x, y, z are the lengths of the medians through A, B, C of a triangle ABC,

    x2 + y2+ z2= (a2+ b2+ c2).

    4.Median always divides a triangle into two equal portions.

    B. Circumcentre: The point of intersection of perpendicular bisectors of the sides of a triangle

  • 7/24/2019 Geometry Capsule

    4/13

    1.The circumcentre is equidistant from the vertices.

    2.If a, b, c, are the sides of the triangle, is the area &R is the radius of the circum-circle, then abc = 4R. 3.In a right angled, the median to the hypotenuse is equal to its circumradius and is equal to half the hypotenuse.

    Orthocentre:The point of intersection of the altitudes of a triangle.

    1. B, Z, Y, C lie on a circle and form a cyclic quadrilateral.

    2.C is the orthocentre of the right angled triangle ABC right angled at C.

    3.Centroid divides the line joining the orthocentre and circumcentre in the ratio of 2 : 1

    Incentre: The point of intersection of angle bisectors of the angles

    1.It is equidistant from the sides of the triangle.

    2.According to Angle bisector theorem BM/MC = AB/AC

    Congruency of triangles:

    Two triangles ABC and DEF are said to be congruent, if they are equal in all respects (equal in shape and size).

    The notation for congruency is or If A = D, B = E, C = F

    AB = DE, BC = EF, AC = DF

    Then ABC DEF or ABC DEF

  • 7/24/2019 Geometry Capsule

    5/13

    Mid-Point Theorem: A line joining the mid points of any two sides of a triangle is parallel and equal to half of the

    third side.

    If in ABC, D & E are the mid points of AB & AC respectively, then we have

    DE || BC

    DE =1/2 BC

    Similar triangles:

    Two figures are said to be similar, if they have the same shape but not the same size.

    NOTE:Congruent triangles are similar but similar triangles need not be congruent.

    Properties of similar triangles:

    If two triangles are similar, the following properties hold true.

    (a)The ratio of the medians is equal to the ratio of the corresponding sides.

    (b)The ratio of the altitudes is equal to the ratio of the corresponding sides.

    (c)The ratio of the internal bisectors is equal to the ratio of corresponding sides.

    (d)The ratio of inradii is equal to the ratio of the corresponding sides.

    (e)The ratio of the circumradii is equal to the ratio of the corresponding sides.

    (f)Ratio of area is equal to the ratio of squares of the corresponding sides.

    (g)Ratio of area is equal to the ratio of squares of the corresponding medians.

    (h)Ratio of area is equal to the ratio of the squares of the corresponding altitudes.

    (i)Ratio of area is equal to the ratio of the squares of the corresponding angle bisectors.

    Basic Proportionality Theorem:

    In a triangle, if a line drawn parallel to one side of a triangle divides the other two sides in the same ratio.

    So if DE is drawn parallel to BC, it would divide sides AB and AC proportionally i.e.

  • 7/24/2019 Geometry Capsule

    6/13

    AD/BD = AE/AC

    Pythagoras Theorem:

    The square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides.

    i.e. in a right angled triangle ABC, right angled at B,

    AC2= AB2+ BC2

    Angle Bisector Theorem:

    If in ABC, CD is the angle bisector ofBCA, the ratio of the lines BD & AD is equal to the ratio of sides containingthe angle.

    BD/AD = BC/ AC

    Circles

    If O is a fixed point in a given plane, the set of points in the plane which are at equal distances from O will form a

    circle.

    Properties of a Circle:

    1. If two chords of a circle are equal, their corresponding arcs have equal measure.

    2.Measurement of an arc is the angle subtended at the centre. Equal arcs subtend equal angles at the center.

    3.A line from centre and perpendicular to a chord bisects the chord.

    4.Equal chords of a circle are equidistant from the centre.

    5. When two circles touch, their centres and their point of contact are collinear.

    6. If the two circles touch externally, the distance between their centres is equal to sum of their radii.

    7.If the two circles touch internally, the distance between the centres is equal to difference of their radii.

    8.Angle at the centre made by an arc is equal to twice the angle made by the arc at any point on the remaining part

    of the circumference.

    Let O be the centre of the circle. BOC = 2 P, when BAC = P

  • 7/24/2019 Geometry Capsule

    7/13

    9.If two chords are equal, the arc containing the chords will also be equal.

    10.There can be one and only one circle that touches three non-collinear points.

    11.The angle inscribed in a semicircle is 90o.

    12.If two chords AB and CD intersect externally at P,

    PA PB = PC PD

    13.If two chords AB and CD intersect internally at P,PA PB = PC PD

    14.If PAB is a secant and PT is a tangent,

    PT2= PA PB

    15.The length of the direct common tangent (PQ)

    Cyclic Quadrilateral

    If a quadrilateral is inscribed in a circle i.e. all the vertex lies on the circumference of the circle, it is said to be a

    cyclic quadrilateral.1. In a cyclic quadrilateral, opposite angles are supplementary.

    2. In a cyclic quadrilateral, if any one side is extended, the exterior angle so formed is equal to the interior opposite

    angle.

    Alternate angle theorem

    Angles in the alternate segments are equal.

    In the given figure, PAT is tangent to the circle and makes angles PAC & BAT respectively with the chords AB &AC.

    Then, BAT = ACB & ABC = PAC

  • 7/24/2019 Geometry Capsule

    8/13

    Polygons

    A closed plane figure made up of several line segments that are joined together is called a polygon.

    Types of Polygons

    Equiangular (All angles equal)

    Equilateral (All sides equal)

    Regular (All sides & angles equal

    Properties of Polygon:

    1.Sum of all the exterior angles of any regular polygon is equal to 3600

    2.Each exterior angle of an n sided regular polygon is (3600/ N) degrees.

    3.Each interior angle of an n sided equiangular polygon is ()

    4.Also as each pair of interior angle & exterior angle is linear.

    5.Each interior angle = 1800exterior angle.6.The sum of all the interior angles of n sided polygon is (n 2)1800

    Some Important Short Cuts

    1. In a ABC, if the bisectors of B and C meet at O then

    BOC = 900+

    A

    2. In a ABC, if sides AB and AC are produced to D and E respectively and the bisectors of DBC and ECB

    Intersect at O, then

  • 7/24/2019 Geometry Capsule

    9/13

    BOC = 900 A

    3. In a ABC, If AD is the angle bisector of BAC and AE BC, then

    BAE = 1/2 ( ABC - ACB)

    4. In a ABC, If BC is Produced to D and AE is the Angle bisector of A, then

    ABC and ACD = 2 AEC

    5. In a right angle ABC,B = 900and AC is hypotenuse. The perpendicular BD is dropped on hypotenuse AC

    from right angle vertex B, then

    (i) BD = (AB x BC)/ (AC)

  • 7/24/2019 Geometry Capsule

    10/13

    (ii) AD = AB2/AC

    (iii) CD = BC2/AC

    (iv) 1/BD2= (1/AB2) + (1/BC2)

    Coordinate Geometry

    1. Equation of line parallel to y-axis

    X = b

    For Example: A Student plotted four points on a graph. Find out which point represents the line parallel to y-

    axis.

    a) (3, 5)

    b) (0, 6)

    c) (8, 0)

    d) (-2, -4)

    Solution: Option (C)

    2. Equation of line parallel to x-axis

    Y = a

    For Example: A Student plotted four points on a graph. Find out which point represents the line parallel to x-

    axis.

    a) (3, 5)

    b) (0, 6)

    c) (8, 0)

    d) (-2, -4)

    Solution: Option (B)

    3. Equations of line

    a) Normal equation of line

    ax + by + c = 0

    b) Slope Intercept Form

    y = mx + c Where, m = slope of the line & c = intercept on y-axis

    For Example:What is the slope of the line formed by the equation 5y - 3x - 10 = 0?

    Solution:5y - 3x - 10 = 0, 5y = 3x + 10

    Y = 3/5 x + 2

    Therefore, slope of the line is = 3/5

    c) Intercept Form

    x/A + y/B = 1, Where, A & B are x-intercept & y-intercept respectively

    For Example:Find the area of the triangle formed the line 4x + 3 y 12 = 0, x-axis and y-axis?Solution:Area of triangle is = * x-intercept * y-intercept.

    Equation of line is 4x + 3 y 12 = 04x + 3y = 12,

    4x/12 + 3y/12 = 1

    x/3 + y/4 = 1

    Therefore area of triangle = * 3 * 4 = 6

  • 7/24/2019 Geometry Capsule

    11/13

    d) Trigonometric form of equation of line, ax + by + c = 0

    x cos + y sin = p,Where, cos = -a/ (a2 + b2) , sin = -b/ (a2 + b2) & p = c/(a2 + b2)

    e) Equation of line passing through point (x1, y1) & has a slope m

    y - y1= m (x-x1)

    4. Slope of line = y2- y1/x2- x1 = - coefficient of x/coefficient of y

    5. Angle between two lines

    Tan = (m2m1)/(1+ m1m2) where, m1, m2= slope of the linesNote: If lines are parallel, then tan = 0If lines are perpendicular, then cot = 0

    For Example:If 7x - 4y = 0 and 3x - 11y + 5 = 0 are equation of two lines. Find the acute angle between the lines?

    Solution:First we need to find the slope of both the lines.

    7x - 4y = 0 y = 74x

    Therefore, the slope of the line 7x - 4y = 0 is 74

    Similarly, 3x - 11y + 5 = 0 y = 311x + 511

    Therefore, the slope of the line 3x - 11y + 5 = 0 is = 311

    Now, let the angle between the given lines 7x - 4y = 0 and 3x - 11y + 5 = 0 is Now,

    Tan = (m2m1)/(1+ m1m2) = [(7/4)(3/11)]/[1+(7/4)*(3/11)]= 1Since is acute, hence we take, tan = 1 = tan 45Therefore, = 45Therefore, the required acute angle between the given lines is 45.

    6.

    Equation of two lines parallel to each otherax + by + c1= 0

    ax + by + c2= 0

    Note: Here, coefficient of x & y are same.

    7. Equation of two lines perpendicular to each other

    ax + by + c1= 0

    bx - ay + c2= 0

    Note: Here, coefficient of x & y are opposite & in one equation there is negative sign.

    8. Distance between two points (x1, y1), (x2, y2)

    D = (x2x1)2+ (y2y1)2

    For Example:Find the distance between (-1, 1) and (3, 4).

    Solution: D = (x2x1)2+ (y2y1)2= (3 (-1))2+ (4 1)2= (16 + 9) = 25 = 5

    9. The midpoint of the line formed by (x1, y1), (x2, y2)

    M = (x1+ x2)/2, (y1+ y2)/2

  • 7/24/2019 Geometry Capsule

    12/13

    10. Area of triangle whose coordinates are (x1, y1), (x2, y2), (x3, y3)

    [ x1(y2y3) + x2(y3y1) + x3(y1y2) ]For Example:Find area of triangle whose vertices are (1, 1), (2, 3) and (4, 5).

    Solution:We have (x1, y1) = (1, 1), (x2, y2) = (2, 3) and (x3, y3) = (4, 5)

    Area of Triangle = [ x1(y2y3) + x2(y3y1) + x3(y1y2) ]=1/2 [ (1(35) +2(51) + 4(13)) ]= 1

  • 7/24/2019 Geometry Capsule

    13/13