To determine the length of one leg in a [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle when the hypotenuse is given, follow these steps:
1. Understand the properties of a [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle:
- This type of triangle is an isosceles right triangle, where the two legs are of equal length.
- The relationship between the hypotenuse and each leg in such a triangle is given by the formula:
[tex]\[
\text{Hypotenuse} = \text{Leg} \times \sqrt{2}
\][/tex]
- Therefore, the leg can be found by rearranging this formula:
[tex]\[
\text{Leg} = \frac{\text{Hypotenuse}}{\sqrt{2}}
\][/tex]
2. Substitute the given hypotenuse length into the formula:
- The hypotenuse length is given as 18 cm.
[tex]\[
\text{Leg} = \frac{18 \, \text{cm}}{\sqrt{2}}
\][/tex]
3. Simplify the expression:
- To simplify this expression, we can multiply the numerator and denominator by [tex]\(\sqrt{2}\)[/tex] to rationalize the denominator:
[tex]\[
\text{Leg} = \frac{18 \, \text{cm}}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{18 \sqrt{2} \, \text{cm}}{2} = 9 \sqrt{2} \, \text{cm}
\][/tex]
4. Evaluate the numerical value (if needed):
- This simplifies the calculation to:
[tex]\[
\text{Leg} = 12.727922061357855 \, \text{cm}
\][/tex]
Hence, the length of one leg of the [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle, given a hypotenuse of 18 cm, is [tex]\( 12.727922061357855 \, \text{cm} \)[/tex].
Therefore, the correct answer choices are:
- To the exact form: [tex]\( 9\sqrt{2} \, \text{cm} \)[/tex]
- To the decimal form: [tex]\( 12.727922061357855 \, \text{cm} \)[/tex]
But based on the value given in the question options, the correct form would be:
[tex]\[ 9\sqrt{2} \, \text{cm} \][/tex]
