Similar Triangles and Area Ratios Practice

Two similar triangles have corresponding sides in the ratio 3:5. What is the ratio of their areas?

9:25 (since 3²:5² = 9:25).

The ratio of the areas of two similar triangles is 49:64. What is the ratio of their corresponding side lengths?

7:8 (take the square root of both numbers: √49 : √64 = 7:8).

Triangle PQR is similar to triangle STU. PQ = 5 cm and ST = 10 cm. If the area of triangle PQR is 18 cm², find the area of triangle STU.

72 cm². The side ratio is 5/10 = 1/2. The area ratio is (1/2)² = 1/4. Setting 1/4 = 18/Area(STU) gives Area(STU) = 72 cm².

Triangle ABC has an area of 100 square inches. A smaller similar triangle, DEF, has an area of 36 square inches. If a side of triangle ABC is 15 inches, how long is the corresponding side of triangle DEF?

9 inches. The area ratio is 36/100, which reduces to 9/25. The side ratio is √(9/25) = 3/5. Set up 3/5 = x/15, so 5x = 45, which gives x = 9 inches.

In triangle KLM, a segment parallel to side LM cuts through KL and KM, creating a smaller top triangle KNP. If the length of KN is 2 cm and KL is 6 cm, what fraction of the area of triangle KLM is covered by triangle KNP?

1/9. Triangle KNP is similar to triangle KLM by AA similarity. The side ratio is 2/6 = 1/3. The area ratio is (1/3)² = 1/9.

Referring to problem 5, what fraction of the total area of triangle KLM is the trapezoid NLMP?

8/9. The total area is 1, and the top triangle is 1/9, so the remaining trapezoid area is 1 - 1/9 = 8/9.