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Sagot :
To find the length of one leg of a [tex]\( 45^{\circ}-45^{\circ}-90^{\circ} \)[/tex] triangle given that the hypotenuse is 4 cm, we can use the properties of this special right triangle.
In a [tex]\( 45^{\circ}-45^{\circ}-90^{\circ} \)[/tex] triangle (also known as an isosceles right triangle):
- The legs are of equal length.
- The relationship between the legs and the hypotenuse is given by the formula:
[tex]\[ \text{leg} = \frac{\text{hypotenuse}}{\sqrt{2}} \][/tex]
Given that the hypotenuse is 4 cm, we can now substitute this value into the formula:
[tex]\[ \text{leg} = \frac{4 \text{ cm}}{\sqrt{2}} \][/tex]
To rationalize the denominator, multiply both numerator and denominator by [tex]\(\sqrt{2}\)[/tex]:
[tex]\[ \text{leg} = \frac{4 \sqrt{2} \text{ cm}}{2} \][/tex]
Simplifying this expression:
[tex]\[ \text{leg} = 2 \sqrt{2} \text{ cm} \][/tex]
Therefore, the length of one leg of the triangle is:
[tex]\[ 2 \sqrt{2} \text{ cm} \][/tex]
The correct answer is:
[tex]\[ 2 \sqrt{2} \text{ cm} \][/tex]
In a [tex]\( 45^{\circ}-45^{\circ}-90^{\circ} \)[/tex] triangle (also known as an isosceles right triangle):
- The legs are of equal length.
- The relationship between the legs and the hypotenuse is given by the formula:
[tex]\[ \text{leg} = \frac{\text{hypotenuse}}{\sqrt{2}} \][/tex]
Given that the hypotenuse is 4 cm, we can now substitute this value into the formula:
[tex]\[ \text{leg} = \frac{4 \text{ cm}}{\sqrt{2}} \][/tex]
To rationalize the denominator, multiply both numerator and denominator by [tex]\(\sqrt{2}\)[/tex]:
[tex]\[ \text{leg} = \frac{4 \sqrt{2} \text{ cm}}{2} \][/tex]
Simplifying this expression:
[tex]\[ \text{leg} = 2 \sqrt{2} \text{ cm} \][/tex]
Therefore, the length of one leg of the triangle is:
[tex]\[ 2 \sqrt{2} \text{ cm} \][/tex]
The correct answer is:
[tex]\[ 2 \sqrt{2} \text{ cm} \][/tex]
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