Geometry:
The chapter "Geometry" carries the weightage of \(26\) marks in the board examination. It covers the concepts like Similarity and its Criteria, Thales and Angle Bisector Theorem, Construction of Triangles, Pythagoras Theorem, Circles and Tangents, Congruency Theorem.
| Total marks \(26\) |
Toal marks \(=\) \(26\)
|
To prepare well for the board examination, it is necessary to understand the following concepts clearly.
- Similarity and its criteria - Find the ratio of the corresponding sides, Determine the area, perimeter and altitude of the triangle
- Thales and Angle Bisector Theorem - Prove the theorems, Apply the theorems to solve the problems
- Construction of Triangles - Construct the triangles
- Pythagoras Theorem - Prove the theorems, Apply the theorem to solve the problems
- Circles and Tangents - Construct tangent to the circle
- Congruency Theorem - Apply the theorem to solve the problems
Important Concept (Learning Outcomes) Expected Question Type Concept dealt with Similarity of triangles Sec A, Sec B Basic proportionality theorem, Pythagoras theorem, Angle bisector theorem Sec B, Sec C - Basic proportionality theorem
- Pythagoras theorem
- Angle bisector theorem
Construction of a triangle Sec D Circles and Tangents Sec A, Sec C - Tangent to the circle
- Median of the Triangle
Let us recall the concepts in Geometry:
1. Two figures are said to be congruent if they have the same shape and size. Two figures are similar if they have the same shape but are not necessarily the same size.
2. Two triangles are similar; if (i) their corresponding angles are equal, and (ii) their corresponding sides are in the same ratio (or proportion).
3. If two angles in a triangle and the two angles in the other triangle are equal, then the third angle in both triangles must be equal. Therefore, AA Similarity Criterion is the same as the AAA Similarity Criterion.
4. SAS Similarity Criterion: If one angle of a triangle is equal to one angle of another, and if the corresponding sides, including these angles, are proportional, then the two triangles are similar.
5. SSS Similarity Criterion: If the three sides of a triangle are proportional to the three corresponding sides of another triangle, then the two triangles are similar.
6. A perpendicular line drawn from the vertex of a right-angled triangle divides the triangle into two similar triangles and the original triangle.
7. If two triangles are similar, then the corresponding sides' ratio is equal to the ratio of their corresponding altitudes.
8. If two triangles are similar, then the corresponding sides' ratio is equal to the ratio of the corresponding perimeters' ratio.
9. The ratio of the area of two similar triangles equals the ratio of the squares of their corresponding sides.
10. If two triangles have a common vertex and their bases are on the same straight line, the ratio between their areas is equal to the ratio between the length of their bases.
11. A scale factor is the ratio of similar figures' corresponding sides.
12. Basic proportionality theorem or Thales Theorem :A straight line is drawn parallel to one side of a triangle intersecting the other two sides divides the sides in the same ratio.
13. Angle Bisector Theorem: The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the corresponding sides containing the angle.
14. Converse of Angle Bisector Theorem: If a straight line through one vertex of a triangle divides the opposite side internally in the ratio of the other two sides, then the line bisects the angle internally at the vertex.
15. Pythagoras Theorem: In a right angle triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
16. Types of construction of triangles: (i) The base, vertical angle and median on the base, (ii) The base, vertical angle and altitude on the base, (iii) The base, vertical angle and the point where the bisector of the vertical angle meets the base.
17. If a line touches the given circle at only one point, then it is called tangent to the circle.
18. Alternate segment theorem: If a line touches a circle and from the point of contact a chord is drawn, the angles between the tangent and the chord are respectively equal to the angles in the corresponding alternate segments.