the true statements are A. WZ =[tex]\(\frac{15}{2}\)[/tex] inches.
To solve this problem, we need to find the missing side lengths and angles of quadrilateral WXYZ and then verify the given statements.
Given:
- Quadrilateral QRST is formed using a combination of a rectangle and a triangle.
- The sides of the rectangle are 3 inches and 2 inches.
- The length of one side of the triangle, RS, is √18 inches.
- Two angles of the triangle are 135 degrees and 45 degrees.
Step 1: Find the missing side lengths and angles of quadrilateral WXYZ.
Since WXYZ is similar to QRST, corresponding sides are proportional.
Let's denote the unknown sides of WXYZ as follows:
- WX = 5 inches (given)
- WY = x (unknown)
- XY = y (unknown)
- YZ = z (unknown)
Since WXYZ is similar to QRST, we can set up proportions:
[tex]\[\frac{{WX}}{{QR}} = \frac{{WY}}{{QS}} = \frac{{XY}}{{RT}} = \frac{{YZ}}{{ST}}\][/tex]
Given that WX = 5 inches and QR = 5 inches (as QRST is a rectangle), we have:
[tex]\[\frac{5}{5} = \frac{x}{2} \implies x = 2\][/tex]
Now, let's find the other sides using the ratio:
[tex]\[\frac{{WY}}{{QS}} = \frac{{XY}}{{RT}} = \frac{{YZ}}{{ST}}\][/tex]
Given that QS = 3 inches (as QRST is a rectangle), we have:
[tex]\[\frac{2}{3} = \frac{y}{\sqrt{18}} \implies y = \frac{2\sqrt{18}}{3}\][/tex]
Given that ST = 2 inches (as QRST is a rectangle), we have:
[tex]\[\frac{2}{2} = \frac{z}{\sqrt{18}} \implies z = \sqrt{18}\][/tex]
So, the side lengths of WXYZ are:
- WY = 2 inches
- XY = [tex]\(\frac{2\sqrt{18}}{3}\)[/tex] inches
- YZ = [tex]\(\sqrt{18}\)[/tex] inches
Step 2: Verify the given statements.
A. WZ = [tex]\(\frac{15}{2}\)[/tex] inches
We need to calculate WZ:
WZ = WX - XZ
WZ = 5 - [tex]\(\sqrt{18}\) = \(\frac{15}{2}[/tex] inches (True)
B. YM
We don't have information about YM, so we can't determine its length. This statement is inconclusive.
C. ZY = 25 inches
We calculated YZ = [tex]\(\sqrt{18}\)[/tex] inches, which is not equal to 25 inches. So, this statement is false.
D. M
We don't have information about point M, so we can't make any conclusions about it. This statement is inconclusive.
E. The area of quadrilateral WXYZ is [tex]\(\frac{525}{8}\)[/tex] square inches.
To find the area of quadrilateral WXYZ, we can use Heron's formula since we have all three side lengths:
[tex]\[\text{Area} = \sqrt{s(s - WY)(s - XY)(s - YZ)}\][/tex]
where [tex]\(s = \frac{{WY + XY + YZ}}{2}\)[/tex]
Calculate s:
[tex]\(s = \frac{{2 + \frac{{2\sqrt{18}}{3} + \sqrt{18}}{2}}}{2} = \frac{{2 + \frac{{2\sqrt{18}}{3} + \sqrt{18}}{2}}}{2}\)[/tex]
[tex]\(s = \frac{{4 + 2\sqrt{18} + 3\sqrt{18}}}{6} = \frac{{4 + 5\sqrt{18}}}{6}\)[/tex]
Now, calculate the area:
[tex]\[\text{Area} = \sqrt{\frac{{4 + 5\sqrt{18}}}{6} \left(\frac{{4 + 5\sqrt{18}}}{6} - 2\right) \left(\frac{{4 + 5\sqrt{18}}}{6} - \frac{{2\sqrt{18}}{3}}{2}\right) \left(\frac{{4 + 5\sqrt{18}}}{6} - \sqrt{18}\right)}\][/tex]
This expression simplifies to:
[tex]\[\text{Area} = \sqrt{\frac{{4 + 5\sqrt{18}}}{6} \left(\frac{{4 + 5\sqrt{18}}}{6} - 2\right) \left(\frac{{4 + 5\sqrt{18}}}{6} - \frac{{2\sqrt{18}}{3}}{2}\right) \left(\frac{{4 + 5\sqrt{18}}}{6} - \sqrt{18}\right)}\][/tex]
[tex]\[\text{Area} = \sqrt{\frac{1}{36}(4 + 5\sqrt{18})(4 - 2)(4 - \frac{{2\sqrt{18}}{3}}{2})(4 - \sqrt{18})}\][/tex]
Now, compute the value of the expression.
[tex]\[\text{Area} = \sqrt{\frac{1}{36}(4 + 5\sqrt{18})(2)(4 - \frac{{2\sqrt{18}}{3}}{2})(4 - \sqrt{18})}\][/tex]
[tex]\[\text{Area} = \sqrt{\frac{1}{36}(4 + 5\sqrt{18})(2)(4 - \sqrt{18})(4 - \sqrt{18})}\][/tex]
[tex]\[\text{Area} = \sqrt{\frac{1}{36}(4 + 5\sqrt{18})(2)(4 - 18)}\][/tex]
[tex]\[\text{Area} = \sqrt{\frac{1}{36}(4 + 5\sqrt{18})(2)(-14)}\][/tex]
[tex]\[\text{Area} = \sqrt{\frac{1}{36}(-28 - 70\sqrt{18})}\][/tex]
[tex]\[\text{Area} = \frac{1}{6}\sqrt{-28 - 70\sqrt{18}}\][/tex]
The area is not [tex]\(\frac{525}{8}\)[/tex] square inches. So, this statement is false.
Therefore, the true statements are A. WZ =[tex]\(\frac{15}{2}\)[/tex] inches.
