Certainly! Let's analyze the properties of a 45-45-90 triangle to determine which statement is true:
A 45-45-90 triangle is a special type of right triangle where the two non-hypotenuse sides (legs) are of equal length, and each of the non-right angles measures 45 degrees.
Given any 45-45-90 triangle, if we denote the length of each leg as [tex]\( x \)[/tex], then the length of the hypotenuse (opposite the right angle) can be determined using the Pythagorean Theorem. The Pythagorean Theorem states that:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
For our 45-45-90 triangle, since both legs are equal, we have:
[tex]\[ x^2 + x^2 = c^2 \][/tex]
Simplifying this:
[tex]\[ 2x^2 = c^2 \][/tex]
Solving for [tex]\( c \)[/tex]:
[tex]\[ c = \sqrt{2x^2} \][/tex]
[tex]\[ c = x\sqrt{2} \][/tex]
This shows that the hypotenuse [tex]\( c \)[/tex] is [tex]\( \sqrt{2} \)[/tex] times the length of either leg [tex]\( x \)[/tex].
Now, let’s evaluate the given statements:
- A. The hypotenuse is [tex]\( \sqrt{2} \)[/tex] times as long as either leg.
This is true based on our derivation because we found that [tex]\( c = x\sqrt{2} \)[/tex].
- B. Each leg is [tex]\( \sqrt{2} \)[/tex] times as long as the hypotenuse.
This statement is false because [tex]\( x \)[/tex] would need to equal [tex]\( c \times \frac{1}{\sqrt{2}} \)[/tex], not [tex]\( c \times \sqrt{2} \)[/tex].
- C. The hypotenuse is [tex]\( \sqrt{3} \)[/tex] times as long as either leg.
This statement is false because our derivation shows that [tex]\( c = x \sqrt{2} \)[/tex], not [tex]\( x \sqrt{3} \)[/tex].
- D. Each leg is [tex]\( \sqrt{3} \)[/tex] times as long as the hypotenuse.
This statement is false for the same reason as above because the legs would not be [tex]\( \sqrt{3} \)[/tex] times the hypotenuse.
Therefore, the true statement about a 45-45-90 triangle is:
A. The hypotenuse is [tex]\( \sqrt{2} \)[/tex] times as long as either leg.
