Geometry Std.10

Yoganand Samant, Apr 20, 2018

SSC Maharashtra Geometry.

Table of Contents

  1. Similarity
    1. Base and height of triangle
    2. Properties of the ratio of areas of triangles
    3. BPT
    4. Basic proportionality theorem
    5. Theorem of angle bisector
    6. Theorem of areas of similar triangles.
    7. TESTS FOR SIMILARITY
  2. Pythagoras Theoram
    1. Pythagoras Theoram
  3. Circle
    1. Circle
  4. Geometric Constructions
    1. 1. Constructions of two distinct Similar Triangles.

1.

Similarity

SSC maharashtra chapter 1

  • 1. Base and height of triangle
  • 2. Properties of the ratio of areas of triangles
  • 3. BPT
  • 4. Basic proportionality theorem
  • 5. Theorem of angle bisector
  • 6. Theorem of areas of similar triangles.
  • 7. TESTS FOR SIMILARITY

1.1.

Base and height of triangle

Triangle has three sides. One can consider [color=#ff0000]any side as base[/color] of triangle.[br][b][color=#ff0000]Perpendicular segment drawn to base from it's opposite vertex is the height[/color][/b] of triangle.[br][br][size=150][size=200]Area of triangle =[math]\frac{1}{2}\times Base\times Height[/math][/size][/size]
– Yoganand Samant

1.2.

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1.3.

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1.4.

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1.5.

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1.6.

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1.7.

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2.

Pythagoras Theoram

  • 1. Pythagoras Theoram

2.1.

Pythagoras Theoram

In a right angled triangle, the area of the square on hypotenuse is equal to the sum of the areas of squares on the remaining sides.
Pythagorean Triplet.
[color=#ff0000][b][size=85][size=150]If a, b, c are natural numbers and a>b, then [(a[sup]2[/sup]+b[sup]2[/sup]), (a[sup]2[/sup]-b[sup]2[/sup]), (2ab)] is Pythagorean triplet.[/size][/size][/b] [br][/color](a[sup]2[/sup]+b[sup]2[/sup])[sup]2 [/sup]= (a[sup]2[/sup])[sup]2[/sup] + 2a[sup]2[/sup]b[sup]2[/sup] + (b[sup]2[/sup])[sup]2[/sup] [br](a[sup]2[/sup]+b[sup]2[/sup])[sup]2 [/sup]= a[sup]4[/sup]+ 2a[sup]2[/sup]b[sup]2[/sup] + b[sup]4[/sup]   ............................(I)[br](a[sup]2 [/sup]- b[sup]2[/sup])[sup]2[/sup]= (a[sup]2[/sup])[sup]2 [/sup] - 2a[sup]2[/sup]b[sup]2[/sup] + (b[sup]2[/sup])[sup]2[/sup]  [br](a[sup]2[/sup]- b[sup]2[/sup])[sup]2 [/sup]= a[sup]4 [/sup]- 2a[sup]2[/sup]b[sup]2[/sup] + b[sup]4[/sup]  ............................(II)[br]Subtracting (II) from (I), we get[br](a[sup]2[/sup]+b[sup]2[/sup])[sup]2[/sup] - (a[sup]2[/sup]- b[sup]2[/sup])[sup]2[/sup] = 4a[sup]2[/sup]b[sup]2[br][/sup](a[sup]2[/sup]+b[sup]2[/sup])[sup]2[/sup] - (a[sup]2[/sup]- b[sup]2[/sup])[sup]2[/sup] = (2ab)[sup]2[br][/sup][size=100][size=150](a[sup]2[/sup]+b[sup]2[/sup])[sup]2[/sup] = (a[sup]2[/sup]- b[sup]2[/sup])[sup]2[/sup]+ (2ab)[sup]2[br][/sup]Therefore [(a[sup]2[/sup]+b[sup]2[/sup]) [sup][/sup], (a[sup]2[/sup]- b[sup]2[/sup])[sup] [/sup]and (2ab)] is Pythagorean Triplet. [br]Taking different values of a & b we can find more and more triplets.[sup][/sup][/size][/size][br][sup][/sup]
[b]Property of 30[sup]0[/sup]-60[sup]0[/sup]-90[sup]0 [/sup]Triangle[br][/b]If acute angles of a right angled triangle are 30[sup]0[/sup],60[sup]0[/sup] and 90[sup]0[/sup], then the side opposite 30[sup]0 [/sup]angle is half of the hypotenuse and the side opposite to 60[sup]0[/sup] angle is[math]\frac{\sqrt{3}}{2}[/math] times the hypotenuse.
Property of 30-60-90 type triangle.
[b]Property of 45[sup]0[/sup]-45[sup]0[/sup]-9[sup]0[/sup]triangle[br][/b]If the acute angles of right angled triangles are 45[sup]0[/sup] each, then each of the perpendicular side is[math]\frac{1}{\sqrt{2}}[/math] times the hypotenuse.
Property of 45-45-90 type triangle
Similarity in Right angled triangle.
Theoram of Geometric Mean
Pythagoras Theoram
Converse of Pythagoras Theoram
Apollonius Theoram
– Yoganand Samant

3.

Circle

  • 1. Circle

3.1.

Circle

Circle is the locus of co-planer points which are unidistance from a point in same plane.
Types of circles
Congruent Circles
Concentric Circles
Circles with same centre are concentric circles.
Intersecting and touchig circles
Circles passing through two points
Infinitely many circles can be draw passing through two points.
Tangent and secant
Converse of Tangent Theorem
Tangent Secant Theorem.
Theorem of touching circles
Activity - Enter the measure of arc and observe the type.
Theorem of congruent chord.
Theorem of congruent arcs.
Angles insribed in same arc
Cyclic quadrilateral
Theorem of cyclic quadrilateral
Internal Division of chords
External Division of Chords
– Yoganand Samant

4.

Geometric Constructions

1.Construction of Similar Triangle. 2. Construction of tangent to Circle.

  • 1. 1. Constructions of two distinct Similar Triangles.

4.1.

1. Constructions of two distinct Similar Triangles.

2. Construction of similar triangles having one common vertex. where dimensions of smaller triangle are given.
3. Construction of similar triangles having one common vertex. Where dimensions of larger triangles are given.
4. Construction of tangent to a circle from point on the circle, using centre of the circle.
4.Construction of tangent to the circle from a point on the circle, without using centre.
5. Construction of tangents to the circle from the point outside the circle.
6. Construction of tangents to the circle at the end point of chord without using centre.
– Yoganand Samant

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