Question

1. in each diagram, calculate the indicated side using the pythagorean theorem and find the indicated angle using the tangent ratio.

a) 1.9 cm 4.3 cm
b) 7.8 m 8.2 m
c) 6.5 in 9.4 in

Explanation:

Step1: Apply Pythagorean theorem for part b

For a right - triangle with sides \(a = 7.8\)m and \(b = 8.2\)m, by the Pythagorean theorem \(a^{2}+b^{2}=r^{2}\), where \(r\) is the hypotenuse. So \(r=\sqrt{7.8^{2}+8.2^{2}}=\sqrt{60.84 + 67.24}=\sqrt{128.08}\approx11.32\)m. Let the angle opposite to side \(a\) be \(\theta\). Then \(\tan\theta=\frac{7.8}{8.2}\approx0.9512\), and \(\theta=\tan^{- 1}(0.9512)\approx43.5^{\circ}\)

Step2: Apply Pythagorean theorem for part c

For a right - triangle with sides \(a = 6.5\)in and \(b = 9.4\)in, by the Pythagorean theorem \(a^{2}+b^{2}=f^{2}\), where \(f\) is the hypotenuse. So \(f=\sqrt{6.5^{2}+9.4^{2}}=\sqrt{42.25+88.36}=\sqrt{130.61}\approx11.43\)in. Let the angle opposite to side \(a\) be \(\alpha\). Then \(\tan\alpha=\frac{6.5}{9.4}\approx0.6915\), and \(\alpha=\tan^{-1}(0.6915)\approx34.6^{\circ}\)

Answer:

b) Hypotenuse \(r\approx11.32\)m, angle \(\approx43.5^{\circ}\)
c) Hypotenuse \(f\approx11.43\)in, angle \(\approx34.6^{\circ}\)