Similarity

Q.1 (B)(ii) In the following figure, seg AE \(\perp\) seg BC, seg DF \(\perp\) line BC, AE = 4, DF = 6, then find \(\dfrac{A(\Delta ABC)}{A(\Delta DBC)}\).

Q.2 (B)(i) In the following figure, in \(\Delta PQR\), seg RS bisects \(\angle R\). If PR = 15, QR = 20, PS = 12, then find QS.

Q.3 (A)(i) In the following figure, in \(\Delta ABC\) ray BD bisects \(\angle ABC\), A – D – C, seg ED \(\parallel\) side BC, A – E – B, then to prove that \(\dfrac{AB}{BC} = \dfrac{AE}{EB}\), complete the following activity.

Activity : In \(\Delta ABC\), ray BD bisects \(\angle B\).

\(\therefore \dfrac{AB}{BC} = \dfrac{\text{____}}{\text{____}}\) .............. (I) (Angle bisector theorem)

In \(\Delta ABC\), seg ED \(\parallel\) side BC.

\(\therefore \dfrac{AE}{EB} = \dfrac{\text{____}}{DC}\) .............. (II) ____

\(\therefore \dfrac{AB}{\text{____}} = \dfrac{\text{____}}{EB}\) .............. from (I) and (II)

Q.4 (i) In the following figure, line \(l\) intersects side AB and side AC of \(\Delta ABC\) in points P and Q respectively. Show that \(\dfrac{A(\Delta APQ)}{A(\Delta ABC)} = \dfrac{AP \times AQ}{AB \times AC}\).

Q.1 (B)(1) The ratio of corresponding sides of similar triangles is 3 : 5, then find the ratio of their areas.

Q.2 (B)(1) Find QP using given information in the figure :

Q.3 (A)(1) In the following figure, seg XY || side AC. If 2AX = 3BX and XY = 9, complete the activity to find the value of AC.

Activity :

2AX = 3BX

∴ \(\dfrac{AX}{BX} = \dfrac{3}{2}\)

∴ \(\dfrac{AX + BX}{BX} = \dfrac{____ + ____}{2}\) .......... (By componendo)

\(\dfrac{AB}{BX} = \dfrac{____}{2}\)

Δ BCA ~ Δ BYX ............. (____ Test of similarity)

∴ \(\dfrac{BA}{BX} = \dfrac{AC}{XY}\) .............. (Corresponding sides of similar triangles)

∴ \(\dfrac{5}{____} = \dfrac{AC}{9}\)

∴ AC = ____

Q.4 (3) In Δ ABC, A-X-B and A-Y-C is such that segment XY || side BC. Segment XY divides Δ ABC in two equal areas, then find \(\dfrac{BX}{AB}\).

Q.1 (B)(1) In the following figure Δ ABC, B – D – C and BD = 7, BC = 20, then find \(\dfrac{A(\Delta ABD)}{A(\Delta ABC)}\).

Q.2 (B)(1) In Δ ABC, DE || BC. If DB = 5.4 cm, AD = 1.8 cm, EC = 7.2 cm, then find AE.

Q.3 (A)(1) In the following figure, XY || seg AC. If 2AX = 3BX and XY = 9. Complete the activity to find the value of AC.

Activity :

2AX = 3BX (Given)
∴ \(\dfrac{AX}{BX} = \dfrac{3}{\text{____}}\)
\(\dfrac{AX + BX}{BX} = \dfrac{3 + 2}{2}\) .......... (by componendo)
\(\dfrac{\text{____}}{BX} = \dfrac{5}{2}\) ................ (I)
Now Δ BCA ~ Δ BYX ............. (____ test of similarity)
∴ \(\dfrac{BA}{BX} = \dfrac{AC}{XY}\) .............. (corresponding sides of similar triangles)
∴ \(\dfrac{\text{____}}{\text{____}} = \dfrac{AC}{9}\) .............. from (I)
∴ AC = ____

Q.4 (1) In the following figure DE || BC, then :

(i) If DE = 4 cm, BC = 8 cm, A(Δ ADE) = 25 cm², find A(Δ ABC).
(ii) If DE : BC = 3 : 5, then find A(Δ ADE) : A(□ DBCE).

Q.1 (A)(1) If \(\Delta ABC \sim \Delta DEF\), \(m\angle B = 60^\circ\), then \(m\angle E\) = ....................

Q.1 (B)(2) The ratio of corresponding sides of similar triangles is 3 : 5, then find the ratio of their areas.

Q.2 (A)(1) In the given figure, AR ⊥ BC, AR ⊥ PQ, then complete the activity for finding \(\dfrac{A(\Delta ABC)}{A(\Delta APQ)}\).

Activity :

\(\dfrac{A(\Delta ABC)}{A(\Delta APQ)} = \dfrac{____ \times AR}{PQ \times ____}\)

\(\therefore \dfrac{A(\Delta ABC)}{A(\Delta APQ)} = \dfrac{____}{____}\)

Q.2 (B)(1) In the following figure, in \(\Delta LMN\), ray MT bisects \(\angle LMN\).

If LM = 6, MN = 10, TN = 8, then find LT.

Q.4 (1) Δ ABC has sides of length 4 cm, 5 cm and 6 cm while Δ PQR has perimeter of 90 cm. If Δ ABC is similar to Δ PQR, then find the length of corresponding sides of Δ PQR.

Q.1 (A)(1) In \(\Delta ABC\) and \(\Delta PQR\), in a one to one correspondence of vertices, \(\dfrac{AB}{QR} = \dfrac{BC}{PR} = \dfrac{CA}{PQ}\), then ....................... statement is true.

Q.2 (B)(2) In \(\Delta PQR\), seg RS bisects \(\angle PRQ\). If PR = 15, RQ = 20, PS = 12, then find SQ.

Q.3 (A)(1) In the given figure, AB || CD || EF. If AC = 12, CE = 9, BD = 8, then complete the following activity to find DF.

Activity : AB || CD || EF

\(\therefore \dfrac{AC}{\text{____}} = \dfrac{\text{____}}{DF}\) ........... (Property of three parallel lines and their transversal)

\(\therefore \dfrac{12}{\text{____}} = \dfrac{\text{____}}{DF}\)

\(\therefore DF = \dfrac{8 \times 9}{\text{____}}\)

\(\therefore DF =\) ____

Q.3 (B)(4) If \(\Delta PQR \sim \Delta LMN\), \(9 \times A(\Delta PQR) = 16 \times A(\Delta LMN)\) and QR = 20, then find MN.

Q.1 (B)(1) If \(\Delta ABC \sim \Delta PQR\) and \(AB : PQ = 2 : 3\), then find the value of \(\dfrac{A(\Delta ABC)}{A(\Delta PQR)}\).

Q.3 (A)(1)

In Δ ABC, ray BD bisects ∠ABC, A – D – C, seg DE || side BC, A – E – B, then for showing \(\dfrac{AB}{BC} = \dfrac{AE}{EB}\), complete the following activity :

Proof :

In Δ ABC, ray BD bisects ∠B

∴ \(\dfrac{____}{BC} = \dfrac{AD}{DC}\) ............ (I) (____)

In Δ ABC, DE || BC

∴ \(\dfrac{____}{EB} = \dfrac{AD}{DC}\) ............ (II) (____)

\(\dfrac{AB}{____} = \dfrac{____}{EB}\) ............ [from (I) and (II)]

Q.3 (B)(4) Prove that, ‘If a line parallel to a side of a triangle intersects the remaining sides in two distinct points, then the line divides the side in the same proportion’.

Q.5 (1) □ ABCD is a trapezium, AB || CD, diagonals of trapezium intersects in point P.

Write the answers of the following questions :

(a) Draw the figure using given information.

(b) Write any one pair of alternate angles and opposite angles.

(c) Write the names of similar triangles with test of similarity.