Question

alonzo and kira are standing on a riverbank, 140 meters apart, at points a and b respectively. (see the figure below.) they want to know the distance from kira to a house located across the river at point c. alonzo measures angle a (angle bac) to be 54°, and kira measures angle b (angle abc) to be 70°. what is the distance from kira to the house? round your answer to the nearest tenth of a meter.

Explanation:

Step1: Find angle C

In triangle \(ABC\), the sum of angles is \(180^\circ\). So, \(\angle C = 180^\circ - 54^\circ - 70^\circ = 56^\circ\).

Step2: Apply the Law of Sines

The Law of Sines states \(\frac{BC}{\sin A}=\frac{AB}{\sin C}\). We know \(AB = 140\) m, \(\angle A = 54^\circ\), \(\angle C = 56^\circ\). Let \(BC = x\) (distance from Kira to house). Then \(\frac{x}{\sin 54^\circ}=\frac{140}{\sin 56^\circ}\).

Step3: Solve for x

\(x=\frac{140\times\sin 54^\circ}{\sin 56^\circ}\). Calculate \(\sin 54^\circ\approx0.8090\), \(\sin 56^\circ\approx0.8290\). So, \(x=\frac{140\times0.8090}{0.8290}\approx\frac{113.26}{0.8290}\approx136.6\).

Answer:

136.6