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27b. Construct a Triangle Given Its Perimeter, an Angle and the Length of the Altitude to the Side Opposite the Angle

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This Demonstration constructs a triangle given the perimeter , the angle at and the length of the altitude from to the base .

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This construction uses a previous method to construct a triangle given the length of the base , the opposite angle and the length of the altitude to the base.

Construction

1. Draw a triangle given the length of the base , the angle opposite the base and the length of the altitude from .

2. Let be the intersection of and the perpendicular bisector of . Similarly, let be the intersection of and the perpendicular bisector of .

3. Then the triangle satisfies the conditions.

Verification

By step 1, . Also, since is isosceles, . Similarly, . So .

Since , .

Subtracting, .

Simplifying, .

By step 1 and isosceles triangles, is the perimeter of .

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Contributed by: Izidor Hafner (October 2017)
Open content licensed under CC BY-NC-SA

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Details

This construction was taken from [1].

Reference

[1] G. Polya, How to Solve It: A New Aspect of Mathematical Method, 2nd ed., Princeton, NJ: Princeton University Press, 1957.

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