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Difficulty: MediumTriangles and Pythagorean Theorem

The entrance of a camping tent is shaped like an isosceles triangle. The base of the tent resting on the ground is 16 feet16\text{ feet} wide, and the two slanted sides are each 17 feet17\text{ feet} long. What is the maximum height of the tent, in feet, from the ground to the peak?

Answer: 15 feet

Answer

15
By dividing the isosceles triangle in half, we form a right triangle with a hypotenuse of 17 and a base leg of 8. Applying the Pythagorean theorem (82+h2=1728^2 + h^2 = 17^2) gives 64+h2=28964 + h^2 = 289, resulting in h2=225h^2 = 225. The square root of 225 is 15, which is the maximum height of the tent.

Step-by-Step Solution

1
Divide the base of the isosceles triangle in half to find the base of the inner right triangle.
16÷2=8 feet16 \div 2 = 8\text{ feet}
The height (altitude) of an isosceles triangle bisects the base, creating a right angle necessary for the Pythagorean theorem.
2
Set up the Pythagorean theorem equation (a2+b2=c2a^2 + b^2 = c^2).
h2+82=172h^2 + 8^2 = 17^2
The vertical height (hh) and half of the base (88) are the legs of the right triangle, while the slanted side (1717) is the hypotenuse.
3
Square the known values.
h2+64=289h^2 + 64 = 289
Simplifying the terms makes it possible to solve for the unknown.
4
Subtract 64 from both sides to isolate h2h^2.
h2=225h^2 = 225
Standard algebraic process for solving a one-variable equation.
5
Take the square root of both sides to find hh.
h=15h = 15
The inverse operation of squaring gives the final linear dimension of the height.

Key Concept

Using the properties of isosceles triangles alongside the Pythagorean theorem to find an unknown altitude
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Topics

Triangles and Pythagorean Theorem

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