Question

question 1.
examine the given information in the diagram.
which two statements are true about the measures of the missing side lengths?
a. $x = 8$
b. $y = 16$
c. $y = 8$
d. $y = 8\sqrt{2}$
e. $x = 8\sqrt{2}$

Explanation:

The given triangle is a right - angled triangle with one angle equal to \(45^{\circ}\). So, it is a \(45 - 45-90\) triangle (an isosceles right - triangle). In a \(45 - 45-90\) triangle, the legs are equal in length and the hypotenuse \(h\) is related to the length of each leg \(l\) by the formula \(h = l\sqrt{2}\).

Step 1: Analyze the legs of the triangle

In the given right - angled triangle, one of the legs (the side with length \(8\)) and the other leg \(x\) are equal because the non - right angle is \(45^{\circ}\). So, \(x = 8\) (since in a \(45 - 45-90\) triangle, the two legs are congruent).

Step 2: Analyze the hypotenuse of the triangle

The hypotenuse \(y\) of a \(45 - 45-90\) triangle with leg length \(l\) is given by the formula \(y=l\sqrt{2}\). Here, \(l = 8\), so \(y = 8\sqrt{2}\).

Answer:

A. \(x = 8\)
D. \(y = 8\sqrt{2}\)