Question

question 29 (1 point)
the hypotenuse, c, of right \\( \triangle a b c \\) is 5.0 cm long. given the trigonometric ratio \\( \cos a = 0.75 \\) for angle a, what is the area of the triangle to the nearest tenth of a \\( \mathrm { cm } ^ { 2 } \\) ?
\\( \bigcirc \\) a) 8.0 cm
\\( \bigcirc \\) b) 5.4 cm
\\( \bigcirc \\) c) 7.3 cm
\\( \bigcirc \\) d) 6.2 cm

Explanation:

Step1: Find adjacent side \(b\)

In a right - triangle, \(\cos A=\frac{b}{c}\). Given \(c = 5.0\) cm and \(\cos A=0.75\).
So, \(b = c\times\cos A=5\times0.75 = 3.75\) cm.

Step2: Find opposite side \(a\)

Using the Pythagorean theorem \(a=\sqrt{c^{2}-b^{2}}\).
\(a=\sqrt{5^{2}-3.75^{2}}=\sqrt{25 - 14.0625}=\sqrt{10.9375}\approx3.307\) cm.

Step3: Calculate the area of the triangle

The area of a right - triangle \(A=\frac{1}{2}ab\).
\(A=\frac{1}{2}\times3.75\times3.307=\frac{1}{2}\times12.40125\approx6.2\) \(cm^{2}\).

Answer:

D. \(6.2\) \(cm^{2}\)