The ratio of the areas of similar triangles is the square of the ratio of their corresponding sides. (4/8)^2 = 1/4.

The ratio of corresponding sides of similar triangles is equal to the ratio of any two corresponding sides. Therefore, the ratio is 5:10, which simplifies to 1:2.

The third angle = 180 - (50 + 70) = 60 degrees.

The ratio of the areas of similar triangles is the square of the ratio of their corresponding sides. Therefore, (2/3)^2 = 4/9.

A triangle with sides in the ratio 3:4:5 is a right triangle, as it satisfies the Pythagorean theorem.

Circumference = 2πr = 2π(5) = 10π cm.

By the triangle inequality theorem, the length of the third side must be greater than the difference of the other two sides. Therefore, the minimum length = 24 - 7 = 17 cm.

The ratio of the areas of similar triangles is the square of the ratio of their corresponding sides. Therefore, (2/3)^2 = 4/9.

Since 5^2 + 12^2 = 13^2 (25 + 144 = 169), triangle DEF is a right triangle.

Using the Pythagorean theorem, EF = √(DE^2 + DF^2) = √(6^2 + 8^2) = √(36 + 64) = √100 = 10 cm.

By the Pythagorean theorem, 5^2 + 12^2 = 25 + 144 = 169 = 13^2, so triangle JKL is a right triangle.

The cosine of 60 degrees is 0.5.

The sine of 30 degrees is 0.5.