∆MNT ~ ∆QRS. Length of altitude drawn from point T is 5 and length of altitude drawn from point S is 9. Find the ratio (A(ΔMNT))/(A(ΔQRS)).

Answer:– Given: ∆MNT ~ ∆QRS (Given) ∠M ≅ ∠Q (Corresponding angles of similar triangles)

Step 1: Use the concept of corresponding angles of similar triangles From the given information, we know that ∆MNT is similar to ∆QRS. We can use the corresponding angles to prove that ∆MLT is also similar to ∆QPS:

∠M ≅ ∠Q (Given) ∠MLT ≅ ∠QPS (Each angle is of measure 90°) Therefore, ∆MLT ~ ∆QPS (AA test of similarity)

Step 2: Use the concept of corresponding sides of similar triangles We can use the concept of corresponding sides of similar triangles to find the ratio of MT to QS:

MT/QS = TL/SP (Corresponding sides of similar triangles) From the problem statement, we know that MT/QS = 5/9.

Step 3: Use the theorem of areas of similar triangles We can use the theorem of areas of similar triangles, which states that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Therefore:

(A(∆MNT))/(A(∆QRS)) = MT^2/QS^2 (A(∆MNT))/(A(∆QRS)) = (MT/QS)^2 (A(∆MNT))/(A(∆QRS)) = (5/9)^2 (From Step 2 and given information that ∆MNT ~ ∆QRS) (A(∆MNT))/(A(∆QRS)) = 25/81

Therefore, the ratio of the areas of ∆MNT and ∆QRS is 25/81.