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The hypotenuse of a [tex]45^{\circ}-45^{\circ}-90^{\circ}[/tex] triangle measures 4 cm. What is the length of one leg of the triangle?

A. 2 cm
B. [tex]2 \sqrt{2}[/tex] cm
C. 4 cm
D. [tex]4 \sqrt{2}[/tex] cm


Sagot :

To solve for the length of one leg in a 45°-45°-90° triangle where the hypotenuse measures 4 cm, we can utilize the special properties of this type of triangle.

A 45°-45°-90° triangle, also known as an isosceles right triangle, has sides that follow a specific ratio:
- The legs are of equal length.
- The hypotenuse is [tex]\( \sqrt{2} \)[/tex] times the length of each leg.

Given the hypotenuse ([tex]\(c\)[/tex]) is 4 cm, we can denote the length of each leg as [tex]\(a\)[/tex]. According to the relationship in a 45°-45°-90° triangle, the hypotenuse is [tex]\( \sqrt{2} \)[/tex] times the leg:

[tex]\[ c = a \sqrt{2} \][/tex]

Substitute the given hypotenuse length into the equation:

[tex]\[ 4 = a \sqrt{2} \][/tex]

To find the length of the leg [tex]\(a\)[/tex], solve for [tex]\(a\)[/tex]:

[tex]\[ a = \frac{4}{\sqrt{2}} \][/tex]

Rationalize the denominator by multiplying the numerator and the denominator by [tex]\(\sqrt{2}\)[/tex]:

[tex]\[ a = \frac{4 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} \][/tex]
[tex]\[ a = \frac{4 \sqrt{2}}{2} \][/tex]
[tex]\[ a = 2 \sqrt{2} \][/tex]

Thus, the length of one leg of the triangle is:

[tex]\[ 2 \sqrt{2} \text{ cm} \][/tex]

So the correct answer is:
[tex]\[ \boxed{2 \sqrt{2} \text{ cm}} \][/tex]