To find the leg of an isosceles right triangle when the hypotenuse is given, let's go through the solution step by step.
1. Understand the properties of the isosceles right triangle:
- An isosceles right triangle has two equal legs, and the angle between them is 90 degrees.
- The hypotenuse is opposite the right angle.
- The relation between the hypotenuse [tex]\( h \)[/tex] and each leg [tex]\( a \)[/tex] in an isosceles right triangle is given by the Pythagorean theorem: [tex]\( h^2 = a^2 + a^2 \)[/tex], which simplifies to [tex]\( h = a \sqrt{2} \)[/tex].
2. Write down the given hypotenuse:
- In this problem, the hypotenuse [tex]\( h \)[/tex] is given as [tex]\( 5 \sqrt{6} \)[/tex].
3. Set up the equation using the relationship between the hypotenuse and the leg:
- Given that [tex]\( h = a \sqrt{2} \)[/tex], we can write:
[tex]\[
5 \sqrt{6} = a \sqrt{2}
\][/tex]
4. Solve for the leg [tex]\( a \)[/tex]:
- To isolate [tex]\( a \)[/tex], divide both sides of the equation by [tex]\( \sqrt{2} \)[/tex]:
[tex]\[
a = \frac{5 \sqrt{6}}{\sqrt{2}}
\][/tex]
- Simplify the right-hand side by rationalizing the denominator. Multiply the numerator and the denominator by [tex]\( \sqrt{2} \)[/tex]:
[tex]\[
a = \frac{5 \sqrt{6} \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{5 \sqrt{12}}{2}
\][/tex]
- Notice that [tex]\( \sqrt{12} = 2 \sqrt{3} \)[/tex], so substitute this back into the equation:
[tex]\[
a = \frac{5 \cdot 2 \sqrt{3}}{2} = 5 \sqrt{3}
\][/tex]
5. Compute the numerical value:
- Calculate the value of [tex]\( 5 \sqrt{3} \)[/tex]:
[tex]\[
5 \sqrt{3} \approx 5 \times 1.732 = 8.660
\][/tex]
- The approximate value of the leg [tex]\( a \)[/tex] is [tex]\( 8.660 \)[/tex].
Therefore, the leg of the isosceles right triangle when the hypotenuse is [tex]\( 5 \sqrt{6} \)[/tex] is approximately [tex]\( 8.660 \)[/tex].
