Area of Triangles

The triangle is a right triangle (5-12-13). Area = 0.5 * 5 * 12 = 30. A rectangle with area 30 and width 10 must have length 3 (30/10). The perimeter of the rectangle is 2 * (length + width) = 2 * (10 + 3) = 26.

The area of a triangle is given by the formula (1/2) * base * height. If the base is halved to a/2 while the height b remains constant, the new area becomes (1/2) * (a/2) * b, which is half of the original area.

Use the Pythagorean theorem to find the missing height: sqrt(10^2 - 8^2) = sqrt(100-64) = 6 cm. Then area = 1/2 × base × height = 1/2 × 8 × 6 = 24 cm².

The area of an equilateral triangle with side s is A = (√3/4)s². Substituting s = 20 cm: A = (√3/4)(20)² = (√3/4)(400) = 100√3 square cm. Option A (200√3) would be for a triangle with side 28.28 cm, Option C (50√3) would be for side 14.14 cm, and Option D (400) would be for an equilateral triangle with approximately side 30.3 cm or could be the result of forgetting the √3/4 factor.

For similar triangles, the ratio of areas is the square of the ratio of ANY corresponding linear dimensions - sides, altitudes, medians, or perimeters. However, the most fundamental and commonly stated version uses the sides directly.

Using the triangle area formula Area = (1/2) × base × height, we can solve for base: base = (2 × Area) ÷ height. With Area = 36 cm² and height = 4 cm, base = (2 × 36) ÷ 4 = 72 ÷ 4 = 18 cm, not 16 cm as stated.