Choose whether the given statement is true or false.

Draw an isosceles triangle ABC in which AB = AC = 6 cm and BC = 5. Constructing a triangle PQR similar to ΔABC in which PQ = 8 cm can be justified.

Question

Choose whether the given statement is true or false.

Draw an isosceles triangle ABC in which AB = AC = 6 cm and BC = 5. Constructing a triangle PQR similar to ΔABC in which PQ = 8 cm can be justified.

Accepted Answer

Let us first construct ΔABC step wise.1. Draw a line segment BC of 5cm.2. Now draw its perpendicular bisector by taking an arc of 6cm at B and C and then name the points of intersection of arcs as O, Q and join the points OQ.3. Now we will draw an arc of 6cm from point B. Mark the point of intersection of the arc and OQ line as A. 4. Now join AC and we get isosceles triangle ABC.Justification: The point A will lie on the bisector because it is a point too common to the equal sides AB and AC in the given isosceles triangle. Now we shall construct ΔPQR stepwise.1. Draw any line BX which makes an acute angle with the line BC. 2. Now we will mark four points at equal distances from each other on the line BX i.e. BB1 = B1B2 = B2 B3 = B3B4 3. We join B3C and draw a line B4R which is parallel to B3C. 4. Similarly draw RP || CA.  5. Now we can verify that PB = 8cm. If we rename B as Q we get our required triangle PQR.Justification:It is given that the two triangles are similar which means the sides will be in ratio. We here have taken the ratio   PQ : PA = 8 : 4 = 4 : 3. Now our goal is to make BR = QR and CR 43 times of BC and AC . Suppose from the construction BB1 = B1B2 = B2 B3 = B3B4  = xBC = 3xCR = x CRBC = 13 ∴ BRBC = BC+ CRBC = 1 + CRBC = 1 +13 = 43We then have drawn RP || CA and using Angle-angle-angle (∴ ∠PBC = ∠ABC, ∠ACB = ∠PRB)  similarity we get ΔPBR ∼ ΔABC    So PRCA = 43.Hence, Constructing a triangle PQR similar to ΔABC in which PQ = 8 cm is justifiedSo, option (1) is correct for this question.Hence, the given statement is true.