To solve the question about a 45-45-90 triangle, let’s first understand the properties of this type of triangle.
A 45-45-90 triangle is a special kind of right triangle where the two legs are of equal length. This means that both angles opposite these legs are 45 degrees each.
One important property of a 45-45-90 triangle is the relationship between the lengths of the legs and the hypotenuse. Here’s how we can determine this relationship:
1. Let’s denote the length of each leg as \( a \).
2. Using the Pythagorean theorem for a right triangle, we have:
[tex]\[
\text{(leg)}^2 + \text{(leg)}^2 = \text{(hypotenuse)}^2
\][/tex]
Substituting the values, we get:
[tex]\[
a^2 + a^2 = \text{(hypotenuse)}^2
\][/tex]
3. Simplifying, we get:
[tex]\[
2a^2 = \text{(hypotenuse)}^2
\][/tex]
4. Taking the square root of both sides, we find:
[tex]\[
\sqrt{2a^2} = \text{hypotenuse}
\][/tex]
[tex]\[
\sqrt{2} \cdot a = \text{hypotenuse}
\][/tex]
This tells us that the hypotenuse is \( \sqrt{2} \) times as long as either leg.
Now let’s examine the given choices:
- A. Each leg is \( \sqrt{3} \) times as long as the hypotenuse. (Incorrect, based on the Pythagorean theorem)
- B. The hypotenuse is \( \sqrt{3} \) times as long as either leg. (Incorrect, does not match the relationship we derived)
- C. Each leg is \( \sqrt{2} \) times as long as the hypotenuse. (Incorrect, this is the inverse of the correct relationship)
- D. The hypotenuse is \( \sqrt{2} \) times as long as either leg. (Correct, matches our derived relationship)
Therefore, the correct choice is:
D. The hypotenuse is [tex]\( \sqrt{2} \)[/tex] times as long as either leg.
