QuestionIs it true that in an isosceles right triangle, the hypotenuse is $\sqrt{2}$ times one leg's length? A. Yes B. No C. Maybe D. Sometimes E. Not applicable

Studdy Solution

STEP 1

Assumptions1. We are dealing with an isosceles right triangle. This means that two sides of the triangle are equal in length (the legs), and the triangle contains a right angle (90 degrees). . The sentence is stating that the length of the hypotenuse (the side opposite the right angle) is $\sqrt{}$ times the length of one leg.

STEP 2

In a right triangle, the relationship between the lengths of the sides is given by the Pythagorean theorem, which states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
$c^2 = a^2 + b^2$ where $c$ is the length of the hypotenuse, and $a$ and $b$ are the lengths of the other two sides.

STEP 3

In an isosceles right triangle, the lengths of the two legs are equal. So we can replace $b$ with $a$ in the Pythagorean theorem.
$c^2 = a^2 + a^2$

STEP 4

implify the right side of the equation.
$c^2 =2a^2$

STEP 5

Take the square root of both sides of the equation to solve for $c$ .
$c = \sqrt{2a^2}$

STEP 6

implify the right side of the equation.
$c = a\sqrt{2}$ This shows that the length of the hypotenuse is indeed $\sqrt{2}$ times the length of one leg in an isosceles right triangle.
So, the sentence "In an isosceles right triangle, the length of the hypotenuse is $\sqrt{2}$ times the length of one leg" is true.
The answer is A. Yes.

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