Triangles – important questions for Boards 2027
CBSE Class 10 Maths
If DE ∥ BC in triangle ABC, with D on AB and E on AC, write the proportionality relating AD/AB and AE/AC.
Answer
AD/AB = AE/AC. The Basic Proportionality Theorem gives this relation because DE is parallel to BC.
In triangle PQR, ∠PST = ∠PRQ and the given parallel-line condition yields ∠PST = ∠PQR. What equality of sides follows, and what type of triangle is PQR?
Answer
PQ = PR, so triangle PQR is isosceles. The given equality ∠PST = ∠PRQ, together with ∠PST = ∠PQR from the parallel lines, gives ∠PRQ = ∠PQR; equal angles have equal opposite sides.
State the Basic Proportionality Theorem for a line drawn parallel to one side of a triangle.
Answer
The Basic Proportionality Theorem states that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
State the two conditions that establish similarity of two triangles, and explain how the correspondence of their angles and sides is expressed.
Answer
1. Two triangles are similar when corresponding angles are equal and corresponding sides are in the same ratio (or proportion); triangles with equal corresponding angles are equiangular. 2. For triangles ABC and PQR, with A ↔ P, B ↔ Q and C ↔ R, ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R, and AB/PQ = BC/QR = AC/PR.
In triangle ABC, points D and E lie on AB and AC respectively, and AD/DB = AE/EC. Prove that DE is parallel to BC.
Answer
The converse of the Basic Proportionality Theorem proves the required parallelism. 1. In triangle ABC, D lies on AB, E lies on AC, and AD/DB = AE/EC. Suppose, for contradiction, that DE is not parallel to BC. 2. Through D, draw DE′ parallel to BC, meeting AC at E′. By the Basic Proportionality Theorem, AD/DB = AE′/E′C. 3. Comparing this equality with the given relation gives AE/EC = AE′/E′C. Let AE = x, AE′ = y and AC = L; then EC = L − x and E′C = L − y. 4. Thus x/(L − x) = y/(L − y). Cross-multiplication gives xL − xy = yL − xy, so x = y and E coincides with E′. Therefore, DE coincides with DE′ and is parallel to BC.
State and explain the SAS similarity criterion for triangles, identifying the angle and the two sides that must satisfy its conditions.
Answer locked. Unlock all answers on SmartKalike
Prove that if two triangles are equiangular, the ratios of their corresponding sides are equal.
Answer locked. Unlock all answers on SmartKalike
If the three sides of one triangle are proportional to the three sides of another triangle in corresponding order, which criterion establishes that the triangles are similar? (a) AA (b) SSS (c) SAS (d) RHS
Answer locked. Unlock all answers on SmartKalike
In trapezium ABCD, AB ∥ DC. Points E and F lie on AD and BC, respectively, and EF ∥ AB. State the proportional relationship between AE, ED, BF and FC.
Answer locked. Unlock all answers on SmartKalike
A pole 2 m tall casts a shadow 1.5 m long. At the same time, a tree casts a shadow 9 m long. Assuming the sun’s rays make the same angle with the ground, use similar figures to find the tree’s height.
Answer locked. Unlock all answers on SmartKalike
16 more probable questions in this chapter, every one with a full answer.
Unlock all answers on SmartKalike