Question

show what you know
which of the following best describes the pythagorean theorem?
the area of a triangle is half the base times the height.
the square of the hypotenuse is equal to the sum of the squares of the legs.
the longest side of a triangle is equal to the height.
all sides of a triangle are always equal.

Explanation:

The Pythagorean theorem specifically relates to right - angled triangles. It states that in a right - angled triangle, if \(a\) and \(b\) are the lengths of the legs (the two shorter sides) and \(c\) is the length of the hypotenuse (the longest side opposite the right angle), then \(a^{2}+b^{2}=c^{2}\). This is exactly what the statement “The square of the hypotenuse is equal to the sum of the squares of the legs” describes.

• The formula for the area of a triangle (\(\text{Area}=\frac{1}{2}\times\text{base}\times\text{height}\)) is a different geometric formula and not related to the Pythagorean theorem.
• The statement “The longest side of a triangle is equal to the height” is incorrect in a general sense and has no relation to the Pythagorean theorem. In a right - angled triangle, the height relative to a particular base can be one of the legs, but the hypotenuse (the longest side) is not equal to the height in the general formula for the area of a triangle.
• The statement “All sides of a triangle are always equal” describes an equilateral triangle, which is a special case of a triangle and not related to the Pythagorean theorem (which is for right - angled triangles).

Answer:

The square of the hypotenuse is equal to the sum of the squares of the legs.