Question

triangle nop is similar to triangle qrs. find the measure of side sq. round your answer to the nearest tenth if necessary. figures are not drawn to scale.

Explanation:

Step1: Find the missing side in triangle NOP

In right triangle NOP, we can find the length of NO using the Pythagorean theorem. Let \( NO = x \), \( OP = 18 \), \( NP = 25 \). Then \( x=\sqrt{25^{2}-18^{2}}=\sqrt{625 - 324}=\sqrt{301}\approx17.35\)? Wait, no, wait. Wait, maybe the triangles are similar, so we can use the ratio of corresponding sides. Wait, first, let's check the right triangle NOP: \( OP = 18 \), \( NP = 25 \), so \( NO=\sqrt{25^{2}-18^{2}}=\sqrt{625 - 324}=\sqrt{301}\approx17.35 \)? Wait, no, maybe I got the sides wrong. Wait, the triangle NOP: angle O is right angle? So \( OP \) and \( NO \) are legs, \( NP \) is hypotenuse. So \( NO=\sqrt{25^{2}-18^{2}}=\sqrt{625 - 324}=\sqrt{301}\approx17.35 \). Then triangle QRS is similar to triangle NOP. Let's assume the correspondence: N corresponds to Q, O corresponds to R, P corresponds to S? Wait, the side RQ is 55? Wait, no, the second triangle: R to S is 55? Wait, the problem says triangle NOP is similar to triangle QRS. So let's find the ratio. Wait, maybe the sides: in triangle NOP, the hypotenuse is NP = 25, one leg is OP = 18, the other leg is NO. In triangle QRS, let's say RQ is a leg, RS is the hypotenuse? Wait, no, maybe the correspondence is NOP ~ QRS, so NP corresponds to QS, OP corresponds to RS? Wait, no, let's re - examine.

Wait, maybe the first triangle: NOP, with right angle at O, so sides: OP = 18, NP = 25 (hypotenuse), so NO is the other leg. The second triangle: QRS, right angle at R? So RQ and RS are legs, QS is hypotenuse. Given that RQ is... Wait, the length of RQ is not given, but RS is 55? Wait, no, the problem says "Find the measure of side SQ". So SQ is the hypotenuse of triangle QRS. Since the triangles are similar, the ratio of corresponding sides is equal.

First, find the length of NO in triangle NOP. Using Pythagoras: \( NO=\sqrt{NP^{2}-OP^{2}}=\sqrt{25^{2}-18^{2}}=\sqrt{625 - 324}=\sqrt{301}\approx17.35 \)? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, 25 - 18: 25 squared is 625, 18 squared is 324, 625 - 324 is 301, square root of 301 is about 17.35. But maybe the triangles are similar with the ratio of the hypotenuse and a leg. Wait, maybe NP (hypotenuse of NOP) corresponds to SQ (hypotenuse of QRS), and OP (leg of NOP) corresponds to RQ (leg of QRS), and NO (leg of NOP) corresponds to RS (leg of QRS). Wait, but RS is given as 55? Wait, no, the second triangle: the side from R to S is 55? Wait, the label is R to S is 55? Wait, the figure: in the second triangle, R is at the top, S at the bottom left, Q at the bottom right? So RQ is a vertical side, RS is horizontal? No, maybe the first triangle: NOP, with N at the left, O at the right (right angle), P at the top. So OP is vertical (length 18), NO is horizontal, NP is the hypotenuse (25). The second triangle: QRS, with R at the right (right angle), S at the left, Q at the top. So RQ is vertical, RS is horizontal (length 55), SQ is the hypotenuse. Since the triangles are similar, the ratio of corresponding sides should be equal. So the ratio of OP (vertical leg of NOP) to RQ (vertical leg of QRS) and NO (horizontal leg of NOP) to RS (horizontal leg of QRS) and NP (hypotenuse of NOP) to SQ (hypotenuse of QRS) should be equal.

First, find NO: \( NO=\sqrt{25^{2}-18^{2}}=\sqrt{625 - 324}=\sqrt{301}\approx17.35 \). Wait, but RS is 55? Wait, no, maybe I got the sides reversed. Wait, maybe the horizontal leg of NOP is OP = 18? No, OP is vertical. Wait, maybe the first triangle: NOP, with OP = 18 (vertical leg), NO = x (horizontal leg), NP = 25 (hypotenuse). The second triangle: QRS, with RS = 55 (horizontal leg), RQ = y (vertical leg), SQ = z (hypotenuse). Since they are similar, \( \frac{NO}{RS}=\frac{OP}{RQ}=\frac{NP}{SQ} \). Wait, but we don't know RQ. Wait, maybe the ratio is based on the legs and hypotenuse. Wait, another approach: in triangle NOP, the legs are 18 and \( \sqrt{25^{2}-18^{2}} \), hypotenuse 25. In triangle QRS, one leg is 55? Wait, no, the second triangle has a side of 55. Wait, maybe the 55 is the length of the leg corresponding to NO. Wait, if NO is \( \sqrt{25^{2}-18^{2}}=\sqrt{301}\approx17.35 \), that can't be, because 55 is larger. Wait, I must have messed up the sides. Wait, maybe the first triangle: NP is 25, OP is 18, and NO is the other leg. Wait, no, maybe the triangle is labeled differently. Wait, maybe the first triangle: N to P is 25, O to P is 18, and N to O is the base. So NO is the base, OP is the height, NP is the hypotenuse. Then NO=\( \sqrt{25^{2}-18^{2}} = 17.35\). Then the second triangle: QRS, with RS = 55 (corresponding to NO), and we need to find SQ (corresponding to NP). So the ratio of NO to RS is \( \frac{17.35}{55}\), then SQ = NP * \( \frac{RS}{NO}\). Wait, no, similar triangles have corresponding sides in proportion. So if NOP ~ QRS, then \( \frac{NP}{SQ}=\frac{NO}{RS}=\frac{OP}{RQ} \). Wait, if RS is 55 (corresponding to NO), and NO is \( \sqrt{25^{2}-18^{2}}\approx17.35 \), then the ratio \( \frac{RS}{NO}=\frac{55}{17.35}\approx3.17 \). Then SQ = NP * \( \frac{RS}{NO}=25\times\frac{55}{17.35}\approx25\times3.17\approx79.25 \)? No, that doesn't seem right. Wait, maybe I got the correspondence wrong.

Wait, maybe the first triangle: NOP, with OP = 18 (one leg), NP = 25 (hypotenuse), so the other leg NO=\( \sqrt{25^{2}-18^{2}} = 17.35\). The second triangle: QRS, with RQ = 18 (corresponding to OP), RS = 55 (corresponding to NO), and SQ (corresponding to NP). Then the ratio of OP to RQ is 1, which is not possible. Wait, no, maybe the hypotenuse of NOP is 25, and the hypotenuse of QRS is SQ, and one leg of NOP is 18, and one leg of QRS is 55? No, that can't be. Wait, maybe the triangles are similar with the ratio of the hypotenuse and a leg. Wait, maybe the first triangle: NOP, legs 18 and NO, hypotenuse 25. The second triangle: QRS, legs 55 and RQ, hypotenuse SQ. So if 18 corresponds to 55, then the ratio is \( \frac{55}{18}\), and SQ = 25 * \( \frac{55}{18}\approx25\times3.0556\approx76.4 \). Wait, no, let's do it properly.

Wait, let's re - calculate NO: \( NO=\sqrt{25^{2}-18^{2}}=\sqrt{625 - 324}=\sqrt{301}\approx17.35 \). Now, if triangle NOP ~ triangle QRS, then the ratio of corresponding sides is equal. Let's assume that NO corresponds to QR, OP corresponds to RS, and NP corresponds to QS. Wait, no, maybe the right angle is at O and R. So angle O = angle R = 90 degrees. So NOP ~ QRS by AA similarity (right angle and another angle). So the sides: OP (leg of NOP) corresponds to RS (leg of QRS), NO (leg of NOP) corresponds to RQ (leg of QRS), NP (hypotenuse of NOP) corresponds to QS (hypotenuse of QRS).

We know OP = 18, NP = 25, NO=\( \sqrt{25^{2}-18^{2}}\approx17.35 \). If RS = 55 (corresponding to NO), then the ratio of similarity \( k=\frac{RS}{NO}=\frac{55}{17.35}\approx3.17 \). Then QS (hypotenuse of QRS) = NP * k = 25 * 3.17≈79.25. But this seems off. Wait, maybe I made a mistake in identifying the sides. Wait, maybe the first triangle: NOP, with OP = 18 (vertical leg), NP = 25 (hypotenuse), so horizontal leg NO=\( \sqrt{25^{2}-18^{2}} = 17.35\). The second triangle: QRS, with RQ = 17.35 (corresponding to NO), RS = 55 (corresponding to OP), and SQ (corresponding to NP). Then the ratio \( \frac{RS}{OP}=\frac{55}{18}\approx3.06 \). Then SQ = NP * \( \frac{RS}{OP}=25\times\frac{55}{18}=\frac{1375}{18}\approx76.4 \).

Wait, let's use the correct method. Let's assume that triangle NOP ~ triangle QRS, with N corresponding to Q, O corresponding to R, P corresponding to S. So:

- \( \angle O=\angle R = 90^{\circ} \)
- \( \frac{OP}{RS}=\frac{NO}{RQ}=\frac{NP}{QS} \)

We know \( OP = 18 \), \( NP = 25 \), \( RS = 55 \) (assuming RS corresponds to OP? No, that would be \( \frac{18}{55}\), then \( QS=\frac{25\times55}{18}=\frac{1375}{18}\approx76.4 \)

Yes, that makes sense. So if OP (length 18) in triangle NOP corresponds to RS (length 55) in triangle QRS, then the ratio of similarity \( k = \frac{RS}{OP}=\frac{55}{18} \). Then the hypotenuse NP (length 25) in triangle NOP corresponds to hypotenuse QS (length we need to find) in triangle QRS. So \( QS=NP\times k = 25\times\frac{55}{18}=\frac{1375}{18}\approx76.4 \)

Step2: Calculate the length of SQ

\( SQ=\frac{25\times55}{18}=\frac{1375}{18}\approx76.4 \) (rounded to the nearest tenth)

Answer:

\( \approx76.4 \)