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The Height Test

Part of the Triangles Without Right Angles section of Coddy's Trigonometry journey. Lesson 12 of 44.

Before any arcsin you can count the triangles. With A and b fixed, the side a has to reach from C down to the base. The shortest distance from C to the base is the height h = b×sin(A).

$Triangle ABC with A = 42°, AC = 12 and the height h from C$

For A = 42° and b = 12, h = 12 sin 42°, about 8.03. Then, for an acute A:

a under 8.03: a cannot reach the base, no triangle (the law of sines gives a sine above 1). a equal to h: one right triangle. a between 8.03 and 12: two triangles. a of 12 or more: one triangle, because the second point lands on A or past it.

Challenge

In triangle ABC, A = 40° and AC = 15 cm. Find the height h from C to the base, to two decimal places, then press Check. That height tells you which lengths of BC give two triangles.

Try it yourself

$Use the figure to find h$

Use the figure to find h

Test yourself

This lesson includes a short quiz. Start the lesson to answer it and track your progress.

All lessons in Triangles Without Right Angles

1Across the Lake

Across the Lake Sines Past 90°Cosines Past 90°Recap: Across the River

4Law of Cosines: SAS

The Law of Cosines An Obtuse Angle Between Through the Hill A Hooked Drive Recap: Two Sides, One Angle

7Area of a Triangle

Area With a Sine Area With an Obtuse Angle Working Backwards The Angle From the Area Heron's Formula Recap: A Garden Bed

2Law of Sines: AAS and ASA

Why the Law of Sines Holds Pair Side With Angle The Third Angle First Two Coast Stations Recap: Law of Sines

5Law of Cosines: SSS

Three Sides, One Angle The Largest Angle A Triangular Field Recap: Three Sides

8Bearings and Navigation

Reading Bearings Two Ships From Port Turning Off Course Finding a Course Angle Recap: Ships at Sea

3The Ambiguous Case

Finding an Angle Two TrianglesThe Height TestAn Obtuse Given Angle Radio Coverage Recap: How Many Triangles?

6Which Law to Use

Sines or Cosines?Not Between Three Towns Recap: Pick the Law

9Final Challenges

Final: Across the Canyon Final: Every Triangle Final: A Plot of Land Final: Ship and Plane Final: The Largest Angle