Answer:
[tex]x=4\sqrt{2}\\y=7[/tex]
Step-by-step explanation:
In order to solve this problem, the easiest approach is to draw a line connecting the vertex between the side with a measure of (y) and (3) to the side with the length (x). This side should be parallel to the side with a length of (4).
Since the new side constructed (call this side (b)), is parallel to the side with a length of (4) and is intersected by segments that are perpendicular to the side with a length of (4), the side (b) and (4) must have the same measure. Therefore (b = 4).
The triangle formed with sides (x) and (b) is a (45 - 45 - 90) triangle. Meaning that its angles have the following measure, (45 - 45- 90). This means that its sides follow the following ratio:
angle : side opposite
[tex]45:x\\45:x\\90:x\sqrt{2}[/tex]
Apply this logic to the given situation.
[tex]x=b*\sqrt{2}\\x = 4\sqrt{2}[/tex]
The side (b) divides the side (y) into two parts. Call the part that is a side of the triangle ([tex]y_1[/tex]) the side that is a part of the rectangle is called ([tex]y_2[/tex]). As per the logic used to find the lengths of side (b), one can use similar logic to find the length of side ([tex]y_2[/tex]). The sides ([tex]y_2[/tex]) and (3) are parallel and are intersected by the sides with a measure of (4) and (b), therefore ([tex]y_2[/tex]) and (3) are cognrunet. Thus ([tex]y_2[/tex] = 3).
The base angles converse theorem states that when two angles in a triangle are congruent, the triangle is isosceles, and the sides opposite the congruent angles are also congruent. Thus, one can state the following:
[tex]y_1=b\\y_1=4[/tex]
Now add up the measurement of ([tex]y_2[/tex]) and ([tex]y_1[/tex]) to find the measure of (y):
[tex]y_2+y_1=y\\4+3=y\\7=y[/tex]
