Question

se the diagram to answer the question.

(image of two similar polygons, one with sides 33, 27, 32, 24 (right angle), the other with sides x, 24.75, 13.5, 15.75, 18 (right angle), 24. figures not drawn to scale.)

these polygons are similar.

what is the value of x?

a. 16
b. 20.25
c. 21
d. 24.75
e. 36

Explanation:

Step1: Find similarity ratio

Take corresponding sides: $\frac{18}{27} = \frac{2}{3}$ (or use $\frac{15.75}{24}=\frac{21}{32}$? No, $\frac{15.75}{24}=\frac{63}{96}=\frac{21}{32}$? Wait no, $\frac{18}{27}=\frac{2}{3}$, check $\frac{13.5}{20.25}$ no, wait $\frac{24.75}{33}=\frac{99}{132}=\frac{3}{4}$? Wait no, $\frac{15.75}{24}=\frac{63}{96}=\frac{21}{32}$ no, wait correct pair: 24 (big) and 15.75 (small): $\frac{15.75}{24}=\frac{63}{96}=\frac{21}{32}$ no, wait 32 (big) and 24 (small): $\frac{24}{32}=\frac{3}{4}$. Yes! $\frac{24}{32}=\frac{3}{4}$, check $\frac{18}{27}=\frac{2}{3}$ no, wait $\frac{15.75}{24}=0.65625$, $\frac{24}{33}=0.727$ no, wait $\frac{24.75}{33}=0.75=\frac{3}{4}$. Yes! $\frac{24.75}{33}=\frac{99}{132}=\frac{3}{4}$. $\frac{15.75}{24}=\frac{63}{96}=\frac{21}{32}$ no, wait 24 (big side) and 15.75 (small): $\frac{15.75}{24}=\frac{63}{96}=\frac{21}{32}$ no, wait 32 (big) and 24 (small): $\frac{24}{32}=\frac{3}{4}$. Yes, $\frac{3}{4}$ is the ratio (small/big).

Step2: Set up proportion for $x$

Big side corresponding to $x$ is 27. So $\frac{x}{27}=\frac{3}{4}$? No, wait $\frac{24.75}{33}=\frac{3}{4}$, so small = $\frac{3}{4}$ * big. Wait no, 24.75 is small, 33 is big: $24.75 = 33 * \frac{3}{4}$. Yes, $33*\frac{3}{4}=24.75$. Now, $x$ corresponds to 27 (big side). So $x=27*\frac{3}{4}$? No, wait 24 (small) corresponds to 32 (big): $24=32*\frac{3}{4}$, correct. 18 (small) corresponds to 24 (big): $18=24*\frac{3}{4}$, correct! 15.75 (small) corresponds to 21? No, wait big side 24, small 15.75: $24*\frac{3}{4}=18$, no, I mixed up. Wait 18 (small) and 24 (big): $\frac{18}{24}=\frac{3}{4}$, so small is $\frac{3}{4}$ of big. Then 24 (small) corresponds to big side $24/\frac{3}{4}=32$, which matches. 24.75 (small) corresponds to $24.75/\frac{3}{4}=33$, which matches. 13.5 (small) corresponds to $13.5/\frac{3}{4}=18$? No, big side is 18? No, big figure has 27,33,32,24. Oh right, 13.5 (small) corresponds to 18? No, big figure doesn't have 18. Wait, $x$ is small side, corresponding to big side 27. So $x=27*\frac{3}{4}$? No, $27*\frac{3}{4}=20.25$. Wait but $\frac{x}{27}=\frac{24}{32}=\frac{3}{4}$, so $x=27*\frac{3}{4}=20.25$.

Answer:

B. 20.25