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G-SRT.9 Warmup M3-044-A09-V01

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

Write a triangle area formula from two sides and the included angle

Problem

In the labeled triangle shown, sides \(b\) and \(c\) include angle \(A\). Write the triangle area formula using these measurements.

Standard triangle labeling with angle A between sides b and c, opposite side a, and an auxiliary altitude h. Open full size
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What this problem is really about

Start by locating the vertex where the two named sides meet; that vertex's angle is the included angle required by the area formula. Treat either side as the base and take the sine component of the other as the perpendicular height. This produces one-half the product of the two enclosing sides and the sine of their shared angle.

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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
Write a triangle area formula from two sides and the included angle