Question

find the lengths of a, b, c, and d shown in the figure below if a + b = 22. round your answers to the nearest tenth. note that the figure is not drawn to scale. figure of a right triangle with segments a, b, c, d and a 51° angle, and a side of length 5

Explanation:

Step1: Find \( a \) using cosine

In the right triangle with hypotenuse 5 and angle \( 51^\circ \), \( \cos(51^\circ)=\frac{a}{5} \). So \( a = 5\cos(51^\circ)\approx5\times0.6293\approx3.1 \).

Step2: Find \( b \) using \( a + b = 22 \)

Since \( a + b = 22 \), \( b = 22 - a\approx22 - 3.1 = 18.9 \).

Step3: Find \( c \) using sine

In the right triangle, \( \sin(51^\circ)=\frac{c}{5} \). So \( c = 5\sin(51^\circ)\approx5\times0.7771\approx3.9 \).

Step4: Find \( d \) using Pythagorean theorem

In the left right triangle, \( d=\sqrt{b^{2}+c^{2}}\approx\sqrt{18.9^{2}+3.9^{2}}\approx\sqrt{357.21 + 15.21}=\sqrt{372.42}\approx19.3 \).

Answer:

\( a\approx3.1 \), \( b\approx18.9 \), \( c\approx3.9 \), \( d\approx19.3 \)