In a right triangle, the area equals one-half the product of the lengths of which sides?

In a right triangle, the area equals one-half the product of the lengths of which sides?

• A. The hypotenuse and a leg
• B. The two legs
• C. The hypotenuse and the altitude to the hypotenuse
• D. The base and the height

Answer: (B)

The key idea is that the area formula for any triangle is 1/2 times base times height. In a right triangle, the two legs are perpendicular, so they naturally serve as base and height. Using one leg as the base and the other leg as the height, the area becomes 1/2 times the product of the lengths of the two legs. The hypotenuse paired with a leg isn’t the perpendicular base-height pair, so that product doesn’t generally give the area. The altitude to the hypotenuse works in its own right (area also equals 1/2 times hypotenuse times that altitude), but that altitude isn’t a side. So the correct relationship uses the two legs.

The key idea is that the area formula for any triangle is 1/2 times base times height. In a right triangle, the two legs are perpendicular, so they naturally serve as base and height. Using one leg as the base and the other leg as the height, the area becomes 1/2 times the product of the lengths of the two legs. The hypotenuse paired with a leg isn’t the perpendicular base-height pair, so that product doesn’t generally give the area. The altitude to the hypotenuse works in its own right (area also equals 1/2 times hypotenuse times that altitude), but that altitude isn’t a side. So the correct relationship uses the two legs.