Height of an Isosceles

Part of the Triangles section of Coddy's Geometry journey. Lesson 62 of 71.

In an isosceles triangle, the height drawn from the apex lands on the midpoint of the base. It cuts the triangle into two matching right triangles. In each right triangle, the equal side is the hypotenuse and half the base is one leg. The height is the other leg, so you can find it with the Pythagorean theorem. Take equal sides 13 and base 10. Half the base is 5.

$An isosceles triangle with equal sides 13 and a dashed height h that splits the base into two halves of 5$

The height splits the base into two halves of 5

h = √(13^2 - 5^2)

h = √(169 - 25) = √(144) = 12

The height is 12, which is shorter than the equal side 13, because a leg is always shorter than the hypotenuse.

Use half the base as the leg of the right triangle, not the whole base. In the example the leg is 5, not 10, so the height is √(13^2 - 5^2). If you put 10 under the root instead, the height comes out wrong.