Exercise 31 Page 457

We are asked to compare the altitudes of two similar triangles. The altitudes divide both triangles into two smaller triangles. Recall that corresponding angles of similar triangles are congruent. Therefore, ∠ QPM ≅ ∠ TSR. Let's focus on △ PMQ and △ SRT.

We will follow three steps to obtain the ratio of the altitudes.

1. Show that △ PMQ and △ SRT are similar triangles.
2. Find the scale factor.
3. Find the ratio of the altitudes.

   Let's do it!

   Step 1

   Note that △ PMQ and △ SRT have two pairs of congruent angles.

   

   Therefore, by the Angle-Angle Similarity Postulate, these two triangles are similar.

   Step 2

   We are told that the scale factor of △ PMN and △ SRW is 4:3. This gives the ratio of the lengths of any two corresponding sides. Let's consider the sides PM and RS.

   

   We can write the ratio of the lengths of PM and RS. PM:RS = 4:3 Note that PM and SR are also corresponding sides in △ PMQ and △ SRT, which are similar triangles. Therefore, the scale factor of △ PMQ and △ SRT is also 4:3.

   Step 3

   Since altitudes MQ and RT are corresponding sides of △ PMQ and △ SRT, this means that the ratio of their lengths is also 4:3. MQ:RT = 4:3