Question

similarity: sss and sas
what value of x will make △onm similar to △srq by the sas similarity theorem?
50
20
25
16

Explanation:

Step1: Recall SAS Similarity

For SAS similarity, two sides must be proportional and the included angle equal. So, $\frac{ON}{SR}=\frac{NM}{RQ}$ (wait, correct sides: $\frac{ON}{RQ}=\frac{NM}{SR}$? Wait, let's identify sides. In $\triangle ONM$: sides $ON = 8$, $NM = 10$, included angle at N. In $\triangle SRQ$: sides $RQ = x$, $SR = 20$, included angle at R (equal to angle at N). So proportionality: $\frac{ON}{RQ}=\frac{NM}{SR}$? Wait, no: SAS is two sides around the equal angle. So angle at N (in $\triangle ONM$) and angle at R (in $\triangle SRQ$) are equal. So sides around angle N: $ON = 8$, $NM = 10$. Sides around angle R: $RQ = x$, $SR = 20$. So the ratio should be $\frac{ON}{RQ}=\frac{NM}{SR}$? Wait, no, $\frac{ON}{SR}=\frac{NM}{RQ}$? Wait, let's check: $ON$ corresponds to $SR$, $NM$ corresponds to $RQ$, and angle between them equal. So $\frac{ON}{SR}=\frac{NM}{RQ}$. So $ON = 8$, $SR = 20$, $NM = 10$, $RQ = x$. So $\frac{8}{20}=\frac{10}{x}$? Wait, no, that would be wrong. Wait, maybe $ON$ corresponds to $RQ$, $NM$ corresponds to $SR$. Let's re-express: $\triangle ONM \sim \triangle SRQ$ by SAS. So angle N = angle R. So sides: $ON$ and $NM$ (around N), $SR$ and $RQ$ (around R). So $\frac{ON}{SR}=\frac{NM}{RQ}$. So $ON = 8$, $SR = 20$, $NM = 10$, $RQ = x$. So $\frac{8}{20}=\frac{10}{x}$? No, that gives $8x = 200$ → $x = 25$. Wait, let's check: $\frac{8}{x}=\frac{10}{20}$? Wait, maybe I mixed up. Let's do it correctly: For SAS, the ratio of the two sides forming the included angle must be equal. So in $\triangle ONM$, sides adjacent to angle N: $ON = 8$, $NM = 10$. In $\triangle SRQ$, sides adjacent to angle R: $RQ = x$, $SR = 20$. So $\frac{ON}{RQ}=\frac{NM}{SR}$. So $\frac{8}{x}=\frac{10}{20}$. Cross-multiplying: $10x = 8 \times 20$ → $10x = 160$ → $x = 16$? Wait, no, that's not. Wait, maybe $\frac{ON}{SR}=\frac{NM}{RQ}$. So $\frac{8}{20}=\frac{10}{x}$ → $8x = 200$ → $x = 25$. Ah, that makes sense. Let's verify: $\frac{8}{20} = \frac{2}{5}$, $\frac{10}{25} = \frac{2}{5}$. Yes, so the ratios are equal. So $x = 25$.

Step2: Solve Proportion

Set up the proportion for SAS similarity: $\frac{ON}{SR} = \frac{NM}{RQ}$. Substitute values: $ON = 8$, $SR = 20$, $NM = 10$, $RQ = x$. So $\frac{8}{20} = \frac{10}{x}$. Cross-multiply: $8x = 20 \times 10$ → $8x = 200$ → $x = \frac{200}{8} = 25$.

Answer:

25