Answer :
To solve for the length of one of the legs in a 45-45-90 triangle where the hypotenuse is 10 units, we can follow these steps:
1. Understand the properties of a 45-45-90 triangle:
- In a 45-45-90 triangle, the two legs are of equal length.
- The relationship between the hypotenuse (h) and each leg (x) is given by the formula:
[tex]\[ h = x \sqrt{2} \][/tex]
2. Substitute the known value into the formula:
- We are given the hypotenuse [tex]\( h = 10 \)[/tex] units.
- Substitute [tex]\( h \)[/tex] into the equation:
[tex]\[ 10 = x \sqrt{2} \][/tex]
3. Solve for [tex]\( x \)[/tex]:
- To isolate [tex]\( x \)[/tex], divide both sides of the equation by [tex]\( \sqrt{2} \)[/tex]:
[tex]\[ x = \frac{10}{\sqrt{2}} \][/tex]
4. Simplify the expression:
- Rationalize the denominator by multiplying the numerator and the denominator by [tex]\( \sqrt{2} \)[/tex]:
[tex]\[ x = \frac{10 \sqrt{2}}{2} \][/tex]
- Simplify the fraction:
[tex]\[ x = 5 \sqrt{2} \][/tex]
Therefore, the length of one of the legs of the triangle is [tex]\( 5 \sqrt{2} \)[/tex] units.
Thus, the correct answer is:
A. [tex]\( 5 \sqrt{2} \)[/tex] units
1. Understand the properties of a 45-45-90 triangle:
- In a 45-45-90 triangle, the two legs are of equal length.
- The relationship between the hypotenuse (h) and each leg (x) is given by the formula:
[tex]\[ h = x \sqrt{2} \][/tex]
2. Substitute the known value into the formula:
- We are given the hypotenuse [tex]\( h = 10 \)[/tex] units.
- Substitute [tex]\( h \)[/tex] into the equation:
[tex]\[ 10 = x \sqrt{2} \][/tex]
3. Solve for [tex]\( x \)[/tex]:
- To isolate [tex]\( x \)[/tex], divide both sides of the equation by [tex]\( \sqrt{2} \)[/tex]:
[tex]\[ x = \frac{10}{\sqrt{2}} \][/tex]
4. Simplify the expression:
- Rationalize the denominator by multiplying the numerator and the denominator by [tex]\( \sqrt{2} \)[/tex]:
[tex]\[ x = \frac{10 \sqrt{2}}{2} \][/tex]
- Simplify the fraction:
[tex]\[ x = 5 \sqrt{2} \][/tex]
Therefore, the length of one of the legs of the triangle is [tex]\( 5 \sqrt{2} \)[/tex] units.
Thus, the correct answer is:
A. [tex]\( 5 \sqrt{2} \)[/tex] units
