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Special Right Triangles Quiz

What is the extended ratio relating the side lengths of a 45-45-90 triangle?

A. [tex]$x: x \sqrt{3}: 2 x$[/tex]
B. [tex]$x: x: x \sqrt{3}$[/tex]
C. [tex][tex]$x: x: x \sqrt{2}$[/tex][/tex]
D. [tex]$x: x \sqrt{2}: 3 x$[/tex]


Sagot :

To determine the extended ratio relating the side lengths of a 45-45-90 triangle, we need to consider the properties of this specific type of triangle.

A 45-45-90 triangle is an isosceles right triangle, meaning it has two equal sides and one right angle (90 degrees). The angles in this triangle are 45 degrees, 45 degrees, and 90 degrees.

Here's how the side lengths relate to each other:

1. The legs adjacent to the 45-degree angles are equal.
2. The hypotenuse, which is opposite the 90-degree angle, is longer and can be found using the Pythagorean Theorem. For a 45-45-90 triangle, if each leg has length [tex]\( x \)[/tex], we can call the hypotenuse length [tex]\( y \)[/tex].

The Pythagorean Theorem states:
[tex]\[ x^2 + x^2 = y^2 \][/tex]

This simplifies to:
[tex]\[ 2x^2 = y^2 \][/tex]

Taking the square root of both sides, we obtain:
[tex]\[ y = x\sqrt{2} \][/tex]

Therefore, the side lengths of a 45-45-90 triangle are in the ratio [tex]\( x : x : x\sqrt{2} \)[/tex].

Given the options:

A. [tex]\( x : x\sqrt{3} : 2x \)[/tex]
B. [tex]\( x : x : x\sqrt{3} \)[/tex]
C. [tex]\( x : x : x\sqrt{2} \)[/tex]
D. [tex]\( x : x\sqrt{2} : 3x \)[/tex]

The correct answer is [tex]\( x : x : x\sqrt{2} \)[/tex], which corresponds to option C.

So, the extended ratio relating the side lengths of a 45-45-90 triangle is [tex]\( x : x : x\sqrt{2} \)[/tex]. The correct answer is option C.