Question

for each of the following, solve for the unknowns.
a. b.
a. ( x = 24 ) and ( y = 6 sqrt { 3 } )
(type exact answers, using radicals as needed.)
b. ( x = square ) and ( y = square )
(type exact answers, using radicals as needed.)

Explanation:

Step1: Use 45 - 45 - 90 triangle properties

In a 45 - 45 - 90 triangle, the legs are equal. Let's consider the right - angled triangle with hypotenuse \(8\sqrt{2}\). If the legs of a right - angled isosceles triangle (45 - 45 - 90) are \(y\) (the non - hypotenuse sides), and the hypotenuse \(c\) is related to the leg \(a\) (here \(a = y\)) by the formula \(c=a\sqrt{2}\). Given \(c = 8\sqrt{2}\), then \(y\sqrt{2}=8\sqrt{2}\), so \(y = 8\).

Step2: Find the value of \(x\)

Since the large triangle is composed of two congruent 45 - 45 - 90 triangles. The length \(x\) is the sum of the bases of the two smaller 45 - 45 - 90 triangles. Each of the smaller 45 - 45 - 90 triangles has a base equal to \(y\) (because in a 45 - 45 - 90 triangle, the legs are equal). So \(x=y + y\). Since \(y = 8\), then \(x=16\).

Answer:

\(x = 16\) and \(y = 8\)