What is the Hypotenuse of an Isosceles Right Triangle if the Measure of the Leg is 11 cm?

In an isosceles right triangle, the two legs are of equal length, and the angles opposite these legs are both 45 degrees. Given that the measure of each leg is 11 cm, we can use the Pythagorean theorem to find the length of the hypotenuse.

The Pythagorean Theorem for Isosceles Right Triangles

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. For an isosceles right triangle, if the legs are both of length a, then the hypotenuse c can be calculated as follows:

c2 a2 a2 2a2 rArr; c 2a2 a2

Calculating the Hypotenuse

Substituting a 11 cm into the formula, we get:

c 112 ≈ 11times 1.414 ≈ 15.56,text{cm}

Therefore, the hypotenuse of the isosceles right triangle is approximately 15.56 cm.

Understanding the Geometry

It is important to note that in any right triangle, the hypotenuse is the longest side, and opposite the right angle. This property, in the case of an isosceles right triangle, leads to the two legs being of equal length. This is a unique feature of isosceles right triangles, and the relationship between the hypotenuse and the legs can be expressed as c a2 where (a) is the length of the leg.

This relationship holds true in Euclidean geometry, which is the standard geometry taught in most high schools. This formula can be applied to any isosceles right triangle, regardless of the specific length of the legs, as long as they are both equal.

Summary

Conclusively, for an isosceles right triangle with each leg of 11 cm, the hypotenuse will be approximately 15.56 cm. This result is derived from the Pythagorean Theorem and the unique properties of isosceles right triangles.