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G-SRT.9 Warmup M3-044-A04-V01

Derive the triangle area formula A=1/2ab sin(C) using an auxiliary altitude.

Find a missing side using the triangle area formula A = (1/2)ab sin(C)

Problem

The triangle shown has area \(30\), side \(a~=~10\), and included angle \(C~=~30°\). Use the triangle area formula to find \(b\).

Triangle labeled with area 30, side a equal to 10, included angle C equal to 30 degrees, and unknown side b. Open full size
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What this problem is really about

Start with the included-angle area formula and substitute the known area, side, and angle before rearranging. The known side's sine component supplies the perpendicular height, while the unknown side remains the base boundary. Isolate that side by dividing by the full coefficient, then substitute it back to verify the stated area.

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Four variants of this problem type
Curriculum context
Course
Math III
Standard
G-SRT.9
Category
Geometry
Domain
Similarity, Right Triangles, and Trigonometry
Objective
Derive the triangle area formula A=1/2ab sin(C) using an auxiliary altitude.
Problem type
Find a missing side using the triangle area formula A = (1/2)ab sin(C)