Question: In the figure given below, the length of PQ is 12 and the length of PR is 15. The area of right triangle STU is equal to the area of the shaded region. If the ratio of the length of ST to the length of TU is equal to the ratio of the length of PQ to the length of QR, what is the length of TU?

(A) (9√2)/4

(B) 9/2

(C) (9√2)/2

(D) 6√2

(E) 12

Solution: The information given in the question seems to overwhelm us but let’s take it a bit at a time.

“length of PQ is 12 and the length of PR is 15”

PQR is a right triangle such that PQ = 12 and PR = 15. So PQ:PR = 4:5. Recall the 3-4-5 triplet. A multiple triplet of 3-4-5 is 9-12-15. This means QR = 9.

“ratio of the length of ST to the length of TU is equal to the ratio of the length of PQ to the length of QR”

ST/TU = PQ/QR

The ratio of two sides of PQR is equal to the ratio of two sides of STU and the included angle between the sides is same ( = 90). Using SAS, triangles PQR and STU are similar.

“The area of right triangle STU is equal to the area of the shaded region”

Area of triangle PQR = Area of triangle STU + Area of Shaded Region

Since area of triangle STU = area of shaded region, (area of triangle PQR) = 2*(area of triangle STU)

In similar triangles, if the sides are in the ratio k, the areas of the triangles are in the ratio k^2. If the ratio of the areas is given as 2 (i.e. k^2 is 2), the sides must be in the ratio √2 (i.e. k must be √2).Since QR = 9,

TU must be 9/√2. But there is no 9/√2 in the options – in the options the denominators are rationalized. TU = 9/√2 = (9*√2)/(√2*√2) = (9*√2)/2.

Answer (C)

Can you pls explain the highlighted part? I understood the concept of the ratio of similar triangles but not able to apply the concept to this question.