Question

find the perimeter of $\triangle nop$. round your answer to the nearest tenth if necessary. figures are not necessarily drawn to scale.
(image of two triangles: left triangle $\triangle mkl$ with sides 19, 26, 28 and angles 76°, 41°, 63°; right triangle $\triangle nop$ with side 65, 47.5, angle 41°, 76°, 63° and side $x$ to find)

Explanation:

Step1: Identify Similar Triangles

First, check the angles of both triangles. In $\triangle MLK$ (wait, actually $\triangle MLM$? No, the first triangle has angles $76^\circ$, $41^\circ$, and the third angle: $180 - 76 - 41 = 63^\circ$. Wait, the second triangle $\triangle NOP$ has angles $41^\circ$, $76^\circ$, $63^\circ$ (since $180 - 41 - 63 = 76^\circ$). So the triangles are similar by AA similarity (since corresponding angles are equal: $76^\circ$, $41^\circ$, $63^\circ$). So $\triangle MLK$ (wait, the first triangle: let's label it as $\triangle MLK$? Wait, the first triangle has sides 19, 26, 28 and angles $76^\circ$ (at M), $41^\circ$ (at L), $63^\circ$ (at K). The second triangle $\triangle NOP$ has angles $41^\circ$ (at O), $76^\circ$ (at P), $63^\circ$ (at N), and sides: OP = 65, PN = 47.5, and ON = x. Wait, let's find the correspondence. Angle at L is $41^\circ$, angle at O is $41^\circ$; angle at M is $76^\circ$, angle at P is $76^\circ$; angle at K is $63^\circ$, angle at N is $63^\circ$. So the correspondence is: L ↔ O, M ↔ P, K ↔ N. So the sides: in the first triangle, side opposite $63^\circ$ (at K) is ML = 26? Wait, no. Wait, in $\triangle MLK$ (let's correct: first triangle is $\triangle MLK$? Wait, vertices M, L, K. So side MK = 19, ML = 26, LK = 28? Wait, angle at M: $76^\circ$, angle at L: $41^\circ$, angle at K: $63^\circ$. So side opposite angle M ($76^\circ$) is LK = 28; side opposite angle L ($41^\circ$) is MK = 19; side opposite angle K ($63^\circ$) is ML = 26. Now in $\triangle NOP$, angle at O: $41^\circ$, angle at P: $76^\circ$, angle at N: $63^\circ$. So side opposite angle O ($41^\circ$) is PN = 47.5; side opposite angle P ($76^\circ$) is ON = x; side opposite angle N ($63^\circ$) is OP = 65. Wait, but in the first triangle, side opposite $41^\circ$ (angle L) is MK = 19, and in the second triangle, side opposite $41^\circ$ (angle O) is PN = 47.5. So the ratio of similarity is $\frac{47.5}{19} = 2.5$. Let's check: in the first triangle, side opposite $76^\circ$ (angle M) is LK = 28. In the second triangle, side opposite $76^\circ$ (angle P) is ON = x. So $x = 28 \times 2.5 = 70$? Wait, no, wait: first triangle sides: MK = 19 (opposite angle L: $41^\circ$), ML = 26 (opposite angle K: $63^\circ$), LK = 28 (opposite angle M: $76^\circ$). Second triangle: PN = 47.5 (opposite angle O: $41^\circ$) → corresponds to MK = 19. So ratio $r = \frac{47.5}{19} = 2.5$. Then OP = 65 (opposite angle N: $63^\circ$) → corresponds to ML = 26? Wait, no: angle N is $63^\circ$, so side opposite angle N is OP = 65. In first triangle, angle K is $63^\circ$, side opposite is ML = 26. So $65 = 26 \times r$ → $r = \frac{65}{26} = 2.5$. Yes, that matches. Then side opposite angle P ($76^\circ$) in second triangle is ON = x, which corresponds to side LK = 28 in first triangle (since angle M is $76^\circ$, side opposite is LK = 28). So $x = 28 \times 2.5 = 70$. Now, the perimeter of $\triangle NOP$ is OP + PN + ON = 65 + 47.5 + 70. Let's calculate that: 65 + 47.5 = 112.5; 112.5 + 70 = 182.5? Wait, wait, no: wait, OP is 65, PN is 47.5, ON is x = 70. So perimeter is 65 + 47.5 + 70 = 182.5? Wait, but let's check the similarity ratio again. Wait, first triangle: sides 19, 26, 28. Second triangle: sides corresponding to 19 (MK) is PN = 47.5 (since angle L (41°) in first triangle, side MK is opposite angle L? Wait, no, in triangle, side opposite angle A is BC. So in first triangle, angle at L is 41°, so side opposite is MK = 19. In second triangle, angle at O is 41°, so side opposite is PN = 47.5. So ratio is 47.5 / 19 = 2.5. Then angle at K is 63°, side opposite is ML = 26. In second triangle, angle at N is 63°, side opposite is OP = 65. So 65 / 26 = 2.5, which matches. Then angle at M is 76°, side opposite is LK = 28. In second triangle, angle at P is 76°, side opposite is ON = x. So x / 28 = 2.5 → x = 70. So the sides of $\triangle NOP$ are OP = 65, PN = 47.5, ON = 70. So perimeter is 65 + 47.5 + 70 = 182.5. Wait, but let's check again. Wait, maybe I mixed up the correspondence. Let's list the angles:

First triangle (let's call it $\triangle MLK$):

- Angle at M: 76° (opposite side LK = 28)
- Angle at L: 41° (opposite side MK = 19)
- Angle at K: 63° (opposite side ML = 26)

Second triangle ($\triangle NOP$):

- Angle at O: 41° (opposite side PN = 47.5)
- Angle at P: 76° (opposite side ON = x)
- Angle at N: 63° (opposite side OP = 65)

So the correspondence is:

- Angle L (41°) ↔ Angle O (41°) → side opposite L (MK = 19) ↔ side opposite O (PN = 47.5)
- Angle M (76°) ↔ Angle P (76°) → side opposite M (LK = 28) ↔ side opposite P (ON = x)
- Angle K (63°) ↔ Angle N (63°) → side opposite K (ML = 26) ↔ side opposite N (OP = 65)

Thus, the ratio of similarity is $\frac{PN}{MK} = \frac{47.5}{19} = 2.5$, $\frac{OP}{ML} = \frac{65}{26} = 2.5$, $\frac{ON}{LK} = \frac{x}{28} = 2.5$ → $x = 70$. Then perimeter of $\triangle NOP$ is $OP + PN + ON = 65 + 47.5 + 70 = 182.5$. Wait, but let's check the first triangle's perimeter: 19 + 26 + 28 = 73. The second triangle's perimeter should be 73 * 2.5 = 182.5, which matches. So that's correct.

Step2: Calculate Perimeter

Now that we have all three sides of $\triangle NOP$: $OP = 65$, $PN = 47.5$, and $ON = 70$, we sum them up to find the perimeter.

Perimeter $P = 65 + 47.5 + 70$

First, add 65 and 47.5: $65 + 47.5 = 112.5$

Then add 70 to that result: $112.5 + 70 = 182.5$

Answer:

The perimeter of $\triangle NOP$ is $\boxed{182.5}$