Which is a true statement about a 45-45-90 triangle?
A. Each leg is 3 times as long as the hypotenuse.
B. Each leg is 12 times as long as the hypotenuse.
C. The hypotenuse is 3 times as long as either leg.
D. The hypotenuse is v2 times as long as either leg.
The correct answer and explanation is:
The correct answer is:
D. The hypotenuse is √2 times as long as either leg.
Explanation:
A 45-45-90 triangle is a special type of isosceles right triangle where the two legs are equal in length, and the angles are 45°, 45°, and 90°. The relationship between the sides of a 45-45-90 triangle can be derived from the properties of right triangles and the Pythagorean theorem.
Let’s assume the length of each leg of the triangle is x. According to the Pythagorean theorem: x2+x2=hypotenuse2x^2 + x^2 = \text{hypotenuse}^2
Simplifying this gives: 2×2=hypotenuse22x^2 = \text{hypotenuse}^2
Taking the square root of both sides: hypotenuse=x2\text{hypotenuse} = x\sqrt{2}
Thus, the length of the hypotenuse is √2 times the length of either leg.
This relationship is key to understanding 45-45-90 triangles. If the legs are of length x, the hypotenuse will always be x√2.
To clarify the options:
- Option A is incorrect because the legs are not three times as long as the hypotenuse.
- Option B is incorrect for the same reason.
- Option C is also incorrect because the hypotenuse is not three times the length of the legs.
- Option D is correct because the hypotenuse is exactly √2 times the length of each leg.
This ratio holds true regardless of the size of the triangle, making the 45-45-90 triangle a useful model for problems involving angles and distances in geometry.