An isosceles right triangle is a triangle with two equal sides and one right angle (90 degrees). Let's analyze the relationship between the lengths of the sides in such a triangle.
1. Definition and Properties:
- Since it is isosceles and right, the two legs adjacent to the right angle are of equal length, let's denote this common length by [tex]\( a \)[/tex].
- The hypotenuse (the side opposite the right angle) is the longest side of the triangle.
2. Using the Pythagorean Theorem:
- In a right triangle, the Pythagorean theorem states that [tex]\( c^2 = a^2 + b^2 \)[/tex], where [tex]\( c \)[/tex] is the hypotenuse and [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the legs.
- For our isosceles right triangle, [tex]\( a = b \)[/tex]. Therefore, the equation becomes:
[tex]\[
c^2 = a^2 + a^2 = 2a^2
\][/tex]
- Solving for [tex]\( c \)[/tex], we get:
[tex]\[
c = \sqrt{2a^2} = a\sqrt{2}
\][/tex]
- Hence, the hypotenuse [tex]\( c \)[/tex] is [tex]\( \sqrt{2} \)[/tex] times as long as either leg [tex]\( a \)[/tex].
3. Verifying the Statements:
- A: "The hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as either leg."
- This is incorrect because we established that the hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long.
- B: "Each leg is [tex]\(\sqrt{3}\)[/tex] times as long as the hypotenuse."
- This is incorrect because the legs are [tex]\( 1 / \sqrt{2} \approx 0.707 \)[/tex] times the hypotenuse, not [tex]\( \sqrt{3} \)[/tex].
- C: "Each leg is [tex]\(\sqrt{2}\)[/tex] times as long as the hypotenuse."
- This is incorrect as well because it is the inverse of the correct relationship.
- D: "The hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as either leg."
- This is the correct statement as we determined.
Conclusion:
The correct statement about an isosceles right triangle is:
D. The hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as either leg.
