Question

what is hj?
\\( \frac { 1 3 \sqrt { 2 } } { 2 } \\)
\\( 1 3 \sqrt { 2 } \\)
\\( \frac { 1 3 \sqrt { 3 } } { 3 } \\)
13

Explanation:

Step1: Identify the triangle type

This is a 45 - 45 - 90 right - triangle. In a 45 - 45 - 90 triangle, the sides are in the ratio $1:1:\sqrt{2}$. Let the legs ( $HG$ and $HJ$) be of length $x$ and the hypotenuse ($GJ$) be of length $c$.

Step2: Apply the 45 - 45 - 90 triangle ratio formula

The formula for the hypotenuse $c$ of a 45 - 45 - 90 triangle is $c = x\sqrt{2}$, where $x$ is the length of each leg. Given $c = 13$, we solve for $x$ (since $HJ=x$). Rearranging the formula $x=\frac{c}{\sqrt{2}}$.

Step3: Rationalize the denominator

Multiply numerator and denominator by $\sqrt{2}$: $x=\frac{13\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{13\sqrt{2}}{2}$.

Answer:

$\frac{13\sqrt{2}}{2}$