Question

triangle fgh is an isosceles right triangle with a hypotenuse that measures 16 units. an altitude, \\(\overline{gj}\\), is drawn from the right angle to the hypotenuse.\
what is the length of \\(\overline{gj}\\)?\
8 units\
2 units\
4 units\
6 units

Explanation:

Step1: Recall properties of isosceles right triangle

In an isosceles right triangle, the legs are equal, and the altitude to the hypotenuse has a special property: it is half the length of the hypotenuse (or equal to the length of the legs, since legs in isosceles right triangle are \( \frac{\text{hypotenuse}}{\sqrt{2}} \), and altitude to hypotenuse is also \( \frac{\text{leg}_1\times\text{leg}_2}{\text{hypotenuse}} \), and since legs are equal, this becomes \( \frac{\text{leg}^2}{\text{hypotenuse}} \), and for isosceles right triangle, leg \( = \frac{\text{hypotenuse}}{\sqrt{2}} \), so substituting, we get \( \frac{(\frac{\text{hypotenuse}}{\sqrt{2}})^2}{\text{hypotenuse}}=\frac{\text{hypotenuse}^2/2}{\text{hypotenuse}}=\frac{\text{hypotenuse}}{2} \)).

Step2: Calculate length of \( \overline{GJ} \)

Given hypotenuse \( = 16 \) units. Then the altitude to the hypotenuse ( \( \overline{GJ} \) ) is \( \frac{16}{2}=8 \) units.

Answer:

8 units (corresponding to the option with "8 units")