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The hypotenuse of a [tex]$45^{\circ}-45^{\circ}-90^{\circ}$[/tex] triangle measures 128 cm. What is the length of one leg of the triangle?

A. 64 cm
B. [tex]$64 \sqrt{2}$[/tex] cm
C. 128 cm
D. [tex]$128 \sqrt{2}$[/tex] cm


Sagot :

To determine the length of one leg of a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle where the hypotenuse is given as 128 cm, we need to understand the properties of such a triangle.

1. Basics of 45-45-90 Triangle:
- In a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle, the legs are of equal length.
- Each leg has a length equal to the hypotenuse divided by the square root of 2.

2. Explanation:
- Let [tex]\( h \)[/tex] be the length of the hypotenuse.

Given:
[tex]\[ h = 128 \text{ cm} \][/tex]

- Let [tex]\( l \)[/tex] be the length of one leg of the triangle.

According to the properties, we have:
[tex]\[ l = \frac{h}{\sqrt{2}} \][/tex]

3. Calculation:
- Substitute the given value of [tex]\( h \)[/tex]:

[tex]\[ l = \frac{128}{\sqrt{2}} \][/tex]

- To simplify [tex]\( l \)[/tex], multiply both the numerator and the denominator by [tex]\(\sqrt{2}\)[/tex]:

[tex]\[ l = \frac{128 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{128 \sqrt{2}}{2} = 64 \sqrt{2} \][/tex]

4. Verification:
Referring to the numerical result obtained:
[tex]\[ l \approx 90.50966799187808 \text{ cm} \][/tex]
Notice that this corresponds closely to [tex]\( 64 \sqrt{2} \)[/tex] cm.

Thus, the length of one leg of the [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle is:
[tex]\[ \boxed{64 \sqrt{2} \text{ cm}} \][/tex]

So, the correct choice from the given options is:
[tex]\[ 64 \sqrt{2} \text{ cm} \][/tex]