Question

are these shapes similar?
(there are two triangles in the image, with side lengths and angles marked. the first triangle has sides 27 mi, 27 mi, 18 mi, and the second has sides 18 mi, 18 mi, 10 mi. there are yes and no options below.)

Explanation:

Step1: Identify triangle types

Triangle \( IJK \): \( IJ = 27 \), \( IK = 27 \), so it's isosceles right? Wait, angles: \( \angle J \) and \( \angle K \) are right? Wait, no, the red angles: \( \angle I \) is equal in both? Wait, triangle \( IJK \): sides \( IJ = 27 \), \( IK = 27 \), \( JK = 18 \). Triangle \( PQR \): \( PR = 18 \), \( PQ = 18 \), \( QR = 10 \)? Wait no, wait the lengths: \( IJK \): \( IJ = 27 \), \( IK = 27 \), \( JK = 18 \). \( PQR \): \( PR = 18 \), \( PQ = 18 \), \( QR = 10 \)? Wait no, maybe check the ratios.

Wait, let's list the sides. For similar triangles, corresponding sides must be in proportion.

Triangle \( IJK \): sides \( IJ = 27 \), \( IK = 27 \), \( JK = 18 \).

Triangle \( PQR \): sides \( PR = 18 \), \( PQ = 18 \), \( QR = 10 \)? Wait no, maybe I misread. Wait the second triangle: \( PR = 18 \), \( PQ = 18 \), \( QR = 10 \)? Wait no, the right angle? Wait, maybe the triangles are isosceles? Wait, no, let's check the ratios.

Wait, \( IJ = 27 \), \( PR = 18 \); \( IK = 27 \), \( PQ = 18 \); \( JK = 18 \), \( QR = 10 \)? Wait that can't be. Wait maybe I made a mistake. Wait the first triangle: \( IJ = 27 \), \( IK = 27 \), \( JK = 18 \). So the two equal sides are 27, base 18. The second triangle: \( PR = 18 \), \( PQ = 18 \), \( QR = 10 \). Wait, no, maybe the angles: the red angles. Let's see, in triangle \( IJK \), \( \angle I \) is a common angle? Wait no, the first triangle has \( IJ = 27 \), \( IK = 27 \), so it's isosceles with \( \angle J = \angle K \)? Wait the red angles: \( \angle I \), \( \angle J \), \( \angle K \). The second triangle: \( \angle P \), \( \angle R \), \( \angle Q \). Wait, maybe the triangles are isosceles, and we check the ratio of corresponding sides.

Wait, let's take the sides:

For triangle \( IJK \): sides are 27, 27, 18.

For triangle \( PQR \): sides are 18, 18, 10.

Wait, no, that can't be. Wait maybe I misread the lengths. Wait the first triangle: \( IJ = 27 \), \( IK = 27 \), \( JK = 18 \). The second triangle: \( PR = 18 \), \( PQ = 18 \), \( QR = 10 \)? Wait, no, maybe the second triangle's sides are 18, 18, and 10? Wait that would mean the ratio of 27/18 = 3/2, and 18/10 = 9/5, which are not equal. Wait, maybe I made a mistake. Wait, maybe the first triangle has \( JK = 18 \), and the second has \( QR = 10 \)? Wait no, maybe the correct sides are:

Wait, first triangle: \( IJ = 27 \), \( IK = 27 \), \( JK = 18 \).

Second triangle: \( PR = 18 \), \( PQ = 18 \), \( QR = 10 \). Wait, no, that's not right. Wait, maybe the first triangle is \( IJ = 27 \), \( IK = 27 \), \( JK = 18 \), and the second is \( PR = 18 \), \( PQ = 18 \), \( QR = 10 \). Then the ratio of 27/18 = 3/2, 27/18 = 3/2, but 18/10 = 9/5, which is not 3/2. Wait, that would mean they are not similar. But wait, maybe I misread the lengths. Wait, maybe the second triangle's QR is 12? No, the problem says 10. Wait, maybe the first triangle's JK is 12? No, the problem says 18. Wait, maybe the triangles are right triangles? Wait, no, the red angles: maybe \( \angle J \) and \( \angle K \) are right angles? Wait, the first triangle: \( IJ = 27 \), \( IK = 27 \), \( JK = 18 \). If \( \angle J \) and \( \angle K \) are right angles, then it's a right isosceles? No, 27, 27, 18: by Pythagoras, 18² + 18² = 648, 27² = 729, not equal. So not right. Wait, maybe the angles: \( \angle I \) is equal in both? Wait the first triangle has \( \angle I \), and the second has \( \angle P \), which are marked equal. Then the two sides around \( \angle I \) are 27 and 27, and around \( \angle P \) are 18 and 18. So the ratio is 27/18 = 3/2. Then the third side: \( JK = 18 \), and the third side of the second triangle should be 18*(2/3) = 12? But the problem says 10. Wait, maybe I misread the length of \( QR \). Wait the problem says 10 mi. So 18 vs 10: 18/10 = 9/5, 27/18 = 3/2. Not equal. So the sides are not in proportion. Wait, but maybe the triangles are isosceles with the equal angles. Wait, no, if the included angle is equal and the sides around it are in proportion, then they are similar (SAS similarity). Wait, in triangle \( IJK \), the two sides around \( \angle I \) are 27 and 27. In triangle \( PQR \), the two sides around \( \angle P \) are 18 and 18. So \( \angle I = \angle P \) (marked), and the sides around the angle are in ratio 27/18 = 3/2. Then the third side should be in ratio 3/2. So \( JK \) should be \( QR * (3/2) \). \( QR = 10 \), so \( 10*(3/2) = 15 \), but \( JK = 18 \). So 18 vs 15: not equal. Wait, maybe I misread the length of \( JK \). Wait the first triangle: \( JK = 18 \) mi? Wait the problem says 18 mi. So 18 vs 15: no. Wait, maybe the second triangle's \( QR \) is 12? No, the problem says 10. Wait, maybe I made a mistake. Wait, let's check again.

First triangle: \( IJ = 27 \), \( IK = 27 \), \( JK = 18 \).

Second triangle: \( PR = 18 \), \( PQ = 18 \), \( QR = 10 \).

Ratio of \( IJ/PR = 27/18 = 3/2 \).

Ratio of \( IK/PQ = 27/18 = 3/2 \).

Ratio of \( JK/QR = 18/10 = 9/5 \).

Since 3/2 ≠ 9/5, the sides are not in proportion. Therefore, the triangles are not similar.

Wait, but maybe the angles: the red angles. Maybe the triangles are isosceles, but the base angles? Wait, no, the included angle is equal, but the sides around it are in ratio 3/2, but the third side is not. So SAS similarity requires the sides around the equal angle to be in proportion. Here, the two sides around \( \angle I \) and \( \angle P \) are in ratio 3/2, but the third side is not, so they are not similar.

Step2: Conclusion

Since the ratios of corresponding sides are not equal (27/18 = 3/2, 18/10 = 9/5), the triangles are not similar.

Answer:

no