Question

calculate the lengths of the 2 unlabeled sides. give the exact answers. type the answers in the boxes below. gj = 6.5 units hj = 6.5√2 units

Explanation:

Step1: Identify the triangle type

The triangle \( \triangle GHJ \) is a right - isosceles triangle (since one angle is \( 90^{\circ} \) and another angle is \( 45^{\circ} \), so the third angle \( \angle J=45^{\circ} \)). In a right - isosceles triangle, the two legs (the sides adjacent to the right angle) are equal in length. The leg \( GH = 6.5 \) units, so the other leg \( GJ \) (the side opposite the \( 45^{\circ} \) angle at \( H \)) is also equal to \( GH \).
So, \( GJ=GH = 6.5 \) units.

Step2: Calculate the hypotenuse \( HJ \)

For a right - isosceles triangle with leg length \( a \), the hypotenuse \( c \) is given by the formula \( c = a\sqrt{2} \) (derived from the Pythagorean theorem \( c^{2}=a^{2}+a^{2}=2a^{2}\), so \( c = \sqrt{2a^{2}}=a\sqrt{2} \)). Here, \( a = 6.5 \), so the hypotenuse \( HJ=6.5\sqrt{2} \) units.

Answer:

\( GJ = 6.5 \) units, \( HJ=6.5\sqrt{2} \) units