A 45-45-90 triangle is a special type of right triangle where the two legs are congruent, meaning they have the same length. This type of triangle is also known for its unique properties regarding the relationship between the lengths of the sides.
Here’s a step-by-step explanation to determine the true statement about a 45-45-90 triangle:
1. Understanding the Angles and Sides of a 45-45-90 Triangle:
- The angles of a 45-45-90 triangle are 45 degrees, 45 degrees, and 90 degrees.
- Let the length of each leg be [tex]\( x \)[/tex].
2. Relationship Between the Legs and the Hypotenuse:
- The hypotenuse is the side opposite the 90-degree angle.
- In a 45-45-90 triangle, the hypotenuse can be derived using the Pythagorean Theorem or by knowing that this special right triangle always follows the ratio [tex]\( 1:1:\sqrt{2} \)[/tex].
- The hypotenuse is [tex]\( \sqrt{2} \)[/tex] times the length of each leg.
3. Matching with the Options Provided:
- Option A: Each leg is [tex]\( \sqrt{2} \)[/tex] times as long as the hypotenuse.
This is incorrect because, in reality, the hypotenuse is [tex]\( \sqrt{2} \)[/tex] times longer than either leg, not the other way around.
- Option B: The hypotenuse is [tex]\( \sqrt{3} \)[/tex] times as long as either leg.
This is incorrect because the factor [tex]\( \sqrt{3} \)[/tex] relates to 30-60-90 triangles, not 45-45-90 triangles.
- Option C: The hypotenuse is [tex]\( \sqrt{2} \)[/tex] times as long as either leg.
This is correct. As discussed, in a 45-45-90 triangle, the hypotenuse equals [tex]\( \sqrt{2} \)[/tex] times the length of either leg.
- Option D: Each leg is [tex]\( \sqrt{3} \)[/tex] times as long as the hypotenuse.
This is incorrect for the same reason as Option A but with the wrong factor; each leg cannot be [tex]\( \sqrt{3} \)[/tex] times longer than the hypotenuse.
Therefore, the true statement about a 45-45-90 triangle is:
C. The hypotenuse is [tex]\( \sqrt{2} \)[/tex] times as long as either leg.
