NCERT Solutions for Class 10 Maths Chapter 6 Triangles
1. What are the criteria while checking the similarity of the triangle?
If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar (AAA similarity criterion).
If in two triangles, two angles of one triangle are respectively equal to the two angles of the other triangle, then the two triangles are similar (AA similarity criterion).
If in two triangles, corresponding sides are in the same ratio, then their corresponding angles are equal and hence the triangles are similar (SSS similarity criterion).
2. What is Pythagoras theorem and what is the converse of it?
Pythagoras theorem is used to prove some activities and solve some problems. You have already learnt the proof of this theorem in 9th grade and this chapter you will learn how to prove the similarity of the triangles using the Pythagorean theorem. The important theorems in this section are: (i) If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.
This above theorem is used to determine the Pythagoras Theorem. The Pythagoras Theorem is stated as “In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.” The Indian mathematician Baudhayan gave out the Pythagoras Theorem. It is also important to know the converse of the theorem. This can be proved in the form of a theorem. The theorem is stated as “In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle”.
3. What is the difference between congruent triangles and similar triangles ?
Congruent triangles are the pair of triangles whose each part is exactly the same as the corresponding part of another triangle. Congruent triangles have the same side length and same angle measurement.
In the above figure, ΔABC and ΔDEF are congruent and thus, they have:
Same side length i.e, AB = DE, BC = EF and AC = DF
Same angle measurement i.e. ∠A = ∠D, ∠B = ∠E and ∠C = ∠F
Congruent triangles are denoted by ΔABC ≅ ΔDEF.
Similar triangles are the pair of triangles that have the same shape and size. Two triangles are said to be similar if their corresponding angles are equal and their corresponding sides are in the same ratio (or proportion).
4. What is Thales' theorem? State and explain its result
Thales theorem states that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.
Thales theorem is also known as basic proportionality theorem.
In the above figure, ABC is a triangle in which a line parallel to side BC intersects other two sides AB and AC at D and E respectively.
Then according to thales theorem :AD/DB = AE/EC.
5. If two similar triangles have equal area, then prove that they are congruent.
Given, two similar triangles (ΔABC ~ ΔXYZ) having equal area.
So, according to the property of area of similar triangles:
ar(ABC) / ar(XYZ) = (AB/XY)² = (BC/YZ)² = (AC/XZ)² …..(1)
It is given that ar(ABC) = ar(XYZ)
So, ar(ABC) / ar(XYZ) = 1
Putting this in equation (1), we obtain:
⇒ (AB/XY)² = (BC/YZ)² = (AC/XZ)² = 1
⇒ AB = XY, BC = YZ and AC = XZ
By SSS congruence criterion: ΔABC ≅ ΔXYZ.