Florida Teacher Certification Examinations (FTCE) Subject Area Practice Test

In an isosceles triangle with an altitude of 6 and equal sides measuring 10, what is the length of the base?

1. 14

2. 12

3. 16

4. 18

The correct answer is: 16

To find the length of the base of the isosceles triangle, we can utilize the properties of right triangles and the Pythagorean theorem. An altitude in an isosceles triangle divides it into two congruent right triangles. Each of these right triangles has one leg as the altitude (6 units), the other leg as half the base, and the hypotenuse as one of the equal sides (10 units). Let’s denote half of the base as \( b/2 \). According to the Pythagorean theorem, we can set up the following equation: \[ \left(\frac{b}{2}\right)^2 + 6^2 = 10^2. \] This can be rewritten as: \[ \left(\frac{b}{2}\right)^2 + 36 = 100. \] Subtract 36 from both sides: \[ \left(\frac{b}{2}\right)^2 = 64. \] Taking the square root of both sides gives: \[ \frac{b}{2} = 8. \] Now, multiplying both sides by 2 yields: \[ b = 16. \] Thus, the length of the