Question

solve the right triangle using the given information. a = 76.5 yd, b = 41.9 yd c = □ yd (simplify your answer. type an integer or a decimal. round to the nearest tenth if needed.) a = □°□ (simplify your answers. type integers. round to the nearest ten minutes if needed ) b = □°□ (simplify your answers. type integers. round to the nearest ten minutes if needed.)

Explanation:

Step1: Find the hypotenuse c using Pythagorean theorem

By the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), where \(a = 76.5\) and \(b=41.9\). So \(c=\sqrt{(76.5)^{2}+(41.9)^{2}}=\sqrt{5852.25 + 1755.61}=\sqrt{7607.86}\approx87.2\) yd.

Step2: Find angle A using tangent function

\(\tan A=\frac{a}{b}\), so \(A=\arctan(\frac{76.5}{41.9})\). \(A=\arctan(1.825776)\approx61.3^{\circ}\). Convert the decimal part to minutes: \(0.3\times60 = 18\) minutes. So \(A = 61^{\circ}18'\).

Step3: Find angle B

Since the sum of angles in a triangle is \(180^{\circ}\) and it's a right - triangle (\(90^{\circ}\) angle), \(B=90^{\circ}-A\). \(B = 90^{\circ}-61^{\circ}18'=28^{\circ}42'\).

Answer:

c = 87.2 yd
A = 61°18'
B = 28°42'