1. **Problem statement:** Aiden and May each place rocks on their side of a river and measure distances to a tree on the opposite bank. We need to determine if their triangles are similar and calculate the river's width using their measurements. 2. **Similarity of triangles:** Triangles are similar if their corresponding angles are equal or their sides are proportional. - For Aiden, the two triangles share the angle at the tree and have a right angle at the river bank (assuming vertical height). The sides along the river bank are 5 m and 12 m, and the total base is 15 m. - For May, the two triangles also share the angle at the tree and have a right angle at the river bank. The sides along the river bank are 8 m and 10 m, with a total base of 18 m (8+10). Thus, both Aiden and May have constructed pairs of similar triangles because they share an angle and have right angles, making the triangles similar by AA (Angle-Angle) criterion. 3. **Using May's triangles to calculate river width:** Let the river width be $w$. May's smaller triangle has base 8 m and height $w$. May's larger triangle has base 18 m (8+10) and height $w + 25$ m (since 25 m is the vertical segment from the second rock to the tree). Using similarity: $$\frac{w}{8} = \frac{w + 25}{18}$$ Cross-multiplied: $$18w = 8(w + 25)$$ $$18w = 8w + 200$$ Subtract $8w$ from both sides: $$18w - 8w = 200$$ $$10w = 200$$ Divide both sides by 10: $$w = \frac{200}{10} = 20$$ So, the river width is 20 m using May's triangles. 4. **Using Aiden's triangles to calculate river width:** Let the river width be $w$. Aiden's smaller triangle has base 5 m and height $w$. Aiden's larger triangle has base 15 m and height $w + 15$ m (since 15 m is the vertical segment from the second rock to the tree). Using similarity: $$\frac{w}{5} = \frac{w + 15}{15}$$ Cross-multiplied: $$15w = 5(w + 15)$$ $$15w = 5w + 75$$ Subtract $5w$ from both sides: $$15w - 5w = 75$$ $$10w = 75$$ Divide both sides by 10: $$w = \frac{75}{10} = 7.5$$ So, the river width is 7.5 m using Aiden's triangles. 5. **Preference of triangles:** May's triangles give a river width of 20 m, Aiden's give 7.5 m. May's measurements involve longer distances and may be more accurate due to less relative measurement error. Therefore, May's triangles are preferred for calculating the river width. **Final answers:** - a) Yes, both constructed pairs of similar triangles by AA similarity. - b) River width using May's triangles: $20$ m. - c) River width using Aiden's triangles: $7.5$ m. - d) Prefer May's triangles due to longer distances and likely better accuracy.