Question

find the lengths of a, b, c, and d shown in the figure below if a + b = 27. round your answers to the nearest tenth. note that the figure is not drawn to scale. figure of a triangle with a right angle, side 7, angle 50°, and segments a, b, c, d as labeled a = b = c = d =

Explanation:

Step1: Find \( a \) using cosine

In the right triangle with hypotenuse 7 and angle \( 50^\circ \), \( \cos(50^\circ)=\frac{a}{7} \). So \( a = 7\cos(50^\circ) \approx 7\times0.6428 \approx 4.5 \).

Step2: Find \( c \) using sine

Using \( \sin(50^\circ)=\frac{c}{7} \), so \( c = 7\sin(50^\circ) \approx 7\times0.7660 \approx 5.4 \).

Step3: Find \( b \) using \( a + b = 27 \)

Since \( a \approx 4.5 \), then \( b = 27 - a \approx 27 - 4.5 = 22.5 \).

Step4: Find \( d \) using Pythagorean theorem

In the left right triangle, \( d=\sqrt{b^2 + c^2} \approx \sqrt{22.5^2 + 5.4^2} \approx \sqrt{506.25 + 29.16} \approx \sqrt{535.41} \approx 23.1 \).

Answer:

\( a \approx 4.5 \), \( b \approx 22.5 \), \( c \approx 5.4 \), \( d \approx 23.1 \)