Math  /  Geometry

Questionx=y=\begin{array}{c} x= \\ y= \end{array} z=z=

Studdy Solution

STEP 1

What is this asking? We need to find the lengths of sides xx, yy, and zz in these two right triangles stuck together! Watch out! Don't mix up your 30-60-90 and 45-45-90 triangle rules!

STEP 2

1. Solve for yy
2. Solve for xx
3. Solve for zz

STEP 3

Alright, let's **start** with the triangle with the **30°** and **90°** angles.
We know the side opposite the 30° angle is **18**.
In a 30-60-90 triangle, the side opposite the 30° angle is *half* the length of the hypotenuse.

STEP 4

So, if 1818 is half the hypotenuse, then the hypotenuse, which is also side yy, must be double that! y=218=36y = 2 \cdot 18 = 36 So, y=36y = \mathbf{36}!

STEP 5

Now, let's look at the other triangle.
We know one angle is **45°** and another is **90°**, making the third angle also **45°**.
This means it's a 45-45-90 triangle, which tells us that the two legs are equal!
Since yy is also a side of this triangle, and we just found that y=36y = \mathbf{36}, we know that xx must also be **36**! x=y=36x = y = 36

STEP 6

Okay, almost there!
We still need to find zz.
Back to our 45-45-90 triangle, the hypotenuse (zz) is 2\sqrt{2} times the length of each leg.
We know x=36x = 36, so zz is xx times 2\sqrt{2}.

STEP 7

z=x2=362z = x \cdot \sqrt{2} = 36\sqrt{2} Therefore, z=362z = \mathbf{36\sqrt{2}}!

STEP 8

We found x=36x = \mathbf{36}, y=36y = \mathbf{36}, and z=362z = \mathbf{36\sqrt{2}}!
We did it!

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