Question

find the areas of the figures for parts (a) and (b) below
(a) area = 496 ft²
(simplify your answer. type an exact answer, using radicals as needed )
(b) area = □ ft²
(simplify your answer. type an exact answer, using radicals as needed )

Explanation:

Step1: Find the base of the triangle in part (b)

Using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = 40\) (hypotenuse) and \(b=30\) (one - leg). Let the other leg (half of the base of the big triangle) be \(a\).

$$a=\sqrt{40^{2}-30^{2}}=\sqrt{(40 + 30)(40 - 30)}=\sqrt{70\times10}=\sqrt{700}=10\sqrt{7}$$

The base of the big triangle \(B = 2a=20\sqrt{7}\)

Step2: Calculate the area of the triangle in part (b)

The area formula for a triangle is \(A=\frac{1}{2}\times base\times height\). Here, base \(B = 20\sqrt{7}\) and height \(h = 30\)

$$A=\frac{1}{2}\times20\sqrt{7}\times30$$

$$A = 300\sqrt{7}$$

Answer:

\(300\sqrt{7}\)