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Difficulty: MediumTriangles and Pythagorean Theorem

An isosceles triangle has two equal sides that each measure 1313 inches and a base that measures 1010 inches. What is the length of the altitude (height) drawn to the base?

  1. A
    88 inches
  2. B
    3636 inches
  3. 1212 inchesAnswer
  4. D
    1818 inches

Answer

The length of the altitude is 1212 inches.
Drawing an altitude to the base of an isosceles triangle splits it into two identical right triangles. The base of 1010 inches is bisected into two 55-inch segments. For one of these right triangles, the base leg is 55 inches and the hypotenuse (the equal side) is 1313 inches. Using the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2), we set up the equation h2+52=132h^2 + 5^2 = 13^2. Solving for h2h^2 gives 169−25=144169 - 25 = 144. Taking the square root of 144144 yields an altitude of 1212 inches.

Step-by-Step Solution

1
Identify the properties of an altitude in an isosceles triangle.
The altitude bisects the 1010-inch base, creating two equal segments of 55 inches each.
This splits the isosceles triangle into two congruent right triangles, allowing the use of the Pythagorean theorem.
2
Set up the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2) for one of the right triangles.
h2+52=132h^2 + 5^2 = 13^2, where hh is the altitude, 55 is the base leg, and 1313 is the hypotenuse.
The equal side of the isosceles triangle serves as the hypotenuse of the newly formed right triangle.
3
Square the known values.
h2+25=169h^2 + 25 = 169.
To begin isolating the unknown altitude squared (h2h^2).
4
Solve for the altitude.
h2=169−25=144h^2 = 169 - 25 = 144, so h=144=12h = \sqrt{144} = 12.
Taking the square root of 144144 gives the final length of the altitude.

Key Concept

Using the Pythagorean theorem to find the altitude of an isosceles triangle
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Topics

Triangles and Pythagorean Theorem

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