The altitude drawn to the base of an isosceles triangle is 12 cm and the…

Concept: In an isosceles triangle, the altitude drawn to the base bisects that base and splits the triangle into two congruent right triangles. In each of them the legs are the altitude and half the base, and the hypotenuse is one of the two equal sides. Two relations therefore govern every such triangle: the Pythagorean relation (equal side)2 = (half of base)2 + (altitude)2, and the area formula Area = 1/2 × base × altitude.

Application: Let each equal side be a cm and the base be b cm, with the altitude to the base equal to 12 cm.

1. Perimeter: 2a + b = 36, so a = 18 − b/2.

2. The altitude bisects the base, so one right triangle has legs b/2 and 12 with hypotenuse a: a2 = (b/2)2 + 122.

3. Substitute a = 18 − b/2: (18 − b/2)2 = (b/2)2 + 144.

4. Expand the left-hand side: 324 − 18b + b2/4 = b2/4 + 144.

5. The b2/4 terms cancel, leaving 324 − 18b = 144, so 18b = 180 and b = 10.

6. Then a = 18 − 10/2 = 13, so the base is 10 cm and each equal side is 13 cm.

7. Area = 1/2 × base × altitude = 1/2 × 10 × 12 = 60 cm2.

Cross-check: the right triangle formed by the altitude has legs 5 cm and 12 cm and hypotenuse 13 cm, the 5-12-13 Pythagorean triple, and the three sides give a perimeter of 13 + 13 + 10 = 36 cm, exactly the value stated in the question.

Hence the area of the triangle is 60 cm2.