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G-SRT.11 Core M3-043-R07-V03

Apply the Laws of Sines and Cosines to find unknown measurements in right and non-right triangles.

Solve a fully specified triangulation model

Problem

\(\text{Two}~\text{observation}~\text{points}\) are \(40~m\) apart on the same side of a tower. The nearer elevation angle is \(35°\), and the farther angle is \(22°\), as shown. Let \(x\) be the nearer horizontal distance; write an equation and find \(x\) and the tower height.

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Subtract tangent values and multiply by the baseline: h=40[tan(35°)-tan(22°)]≈11.8 m.

Use the same distance x in both triangles: x tan(35°)=x tan(22°), giving x=0 and height 0 m.

Let x be the nearer distance. x tan(35°)=(x+40)tan(22°), so x≈54.56 m and the tower height is x tan(35°)≈38.2 m.

Treat the horizontal baseline as vertical height and report h=40 m.

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Curriculum context

Course: Math III
Standard: G-SRT.11
Category: Geometry
Domain: Similarity, Right Triangles, and Trigonometry
Objective: Apply the Laws of Sines and Cosines to find unknown measurements in right and non-right triangles.
Problem type: Solve a fully specified triangulation model