Question

ali and kaitlin are standing on a riverbank, 240 meters apart, at points \\(a\\) and \\(b\\) respectively. (see the figure below.) they want to know the distance from kaitlin to a house located across the river at point \\(c\\). ali measures angle \\(a\\) (angle \\(bac\\)) to be \\(50^{\circ}\\), and kaitlin measures angle \\(b\\) (angle \\(abc\\)) to be \\(60^{\circ}\\). what is the distance from kaitlin to the house? round your answer to the nearest tenth of a meter.

Explanation:

Identify the given values

Using the Solving Triangles knowledge point

• Side \(c = AB = 240\text{ m}\)
• Angle \(A = 50^\circ\)
• Angle \(B = 60^\circ\)
• Target: Distance from Kaitlin (at \(B\)) to the house (at \(C\)), which is side \(a = BC\).

Calculate the third angle

Using the Solving Triangles knowledge point

$$ C = 180^\circ - (50^\circ + 60^\circ) = 70^\circ $$

Apply the Law of Sines

Using the Law of Sines knowledge point

$$ \frac{a}{\sin(A)} = \frac{c}{\sin(C)} $$

$$ a = \frac{240 \cdot \sin(50^\circ)}{\sin(70^\circ)} $$

Compute the final numerical value

Using the Solving Triangles knowledge point

$$ a \approx \frac{240 \cdot 0.76604}{0.93969} \approx 195.646\text{ m} $$

Rounding to the nearest tenth gives \(195.6\text{ m}\).

Answer:

Ali and Kaitlin are standing on a riverbank, 240 meters apart, at points A and B respectively. They want to know the distance from Kaitlin to a house located across the river at point C. Ali measures angle A to be 50 degrees, and Kaitlin measures angle B to be 60 degrees. What is the distance from Kaitlin to the house? Round your answer to the nearest tenth of a meter.

<blank>195.6</blank> meters