Question

diagram shows the intersections of several straight roads. the avenues run parallel to each other.
amana walks along oak from point a to b. to the nearest foot, how far does she walk?
347 ft
226 ft
307 ft
75 ft

Explanation:

Step1: Identify Similar Triangles

First Ave and Second Ave are parallel, so the triangles formed by Main St and Oak are similar. By the Basic Proportionality Theorem (Thales' theorem), the ratios of corresponding sides are equal. Let \( AB = x \), the side we need to find. The sides of the larger triangle: 280 ft (hypotenuse), and the smaller triangle: 140 ft (hypotenuse), 113 ft (base). Wait, actually, the ratio of the hypotenuses is \( \frac{280}{140} = 2 \). But maybe better to set up proportion for the bases and the segments on Oak. Wait, the two triangles (formed by Main St intersecting First Ave, Second Ave, and Oak) are similar. So the ratio of the distances on Main St (280 ft and 140 ft) is 2:1. Therefore, the ratio of the segments on Oak (from A to B and from B to the intersection with the other road, which is 113 ft? Wait, no, let's check the diagram. Wait, the smaller triangle has a base of 113 ft? Wait, no, the diagram shows: Oak has point A, then B, then a segment of 113 ft. The smaller triangle (with Second Ave) has a hypotenuse of 140 ft, and a base of 113 ft? Wait, maybe the sides are proportional. Let's denote \( AB = x \). Then, the ratio of the hypotenuses (280/140) should equal the ratio of the bases (x + 113)/113? Wait, no, maybe the two triangles are similar, so \( \frac{280}{140} = \frac{AB + 113}{113} \)? Wait, no, that doesn't make sense. Wait, maybe the first triangle (with First Ave) has a hypotenuse of 280 ft, and the second (with Second Ave) has 140 ft. So the scale factor is 2. Therefore, the base of the first triangle (on Oak) should be twice the base of the second. The second triangle's base (on Oak) is 113 ft? Wait, no, the second triangle's base is from B to the end, which is 113 ft, and the first triangle's base is from A to that end, which is \( AB + 113 \). So \( \frac{280}{140} = \frac{AB + 113}{113} \). Wait, 280/140 is 2, so \( 2 = \frac{x + 113}{113} \), so \( x + 113 = 226 \), so \( x = 226 - 113 = 113 \)? No, that can't be. Wait, maybe I got the triangles wrong. Alternatively, maybe the two triangles are similar with ratio 280:140 = 2:1, so the corresponding sides on Oak are in ratio 2:1. The smaller triangle has a side of 113 ft, so the larger one would have 226 ft? Wait, no, the question is how far Amana walks from A to B, so AB. Wait, maybe the correct proportion is \( \frac{280}{140} = \frac{AB}{113} \)? Wait, 280/140 is 2, so AB = 2*113 = 226? Wait, that matches one of the options. Let's check again. The two triangles are similar because First Ave || Second Ave, so corresponding angles are equal, so by AA similarity, the triangles are similar. The hypotenuse of the larger triangle is 280 ft, hypotenuse of smaller is 140 ft, so scale factor 2. The base of the smaller triangle (on Oak) is 113 ft, so base of larger triangle (on Oak) is 2*113 = 226 ft. But the larger triangle's base is from A to the end, which is AB + 113? Wait, no, maybe the smaller triangle's base is from B to the end (113 ft), and the larger triangle's base is from A to the end (AB + 113). So AB + 113 = 226, so AB = 226 - 113 = 113? No, that's not matching. Wait, maybe the diagram is such that the segment from B to the intersection with the other road (not Oak) is 113 ft, and the segment from A to B is x, and the hypotenuse of the smaller triangle is 140 ft (from B to the intersection with Main St), and the hypotenuse of the larger triangle is 280 ft (from A to the intersection with Main St). So the two triangles (A to Main St intersection, and B to Main St intersection) are similar. So the ratio of hypotenuses is 280/140 = 2. Therefore, the ratio of the legs (on Oak) should be 2. So the leg of the smaller triangle (from B to the intersection with the other road, which is 113 ft? No, maybe the leg on Oak for the smaller triangle is 113 ft, and for the larger triangle is x (AB) + 113? Wait, no, maybe the leg on Oak for the smaller triangle is 113 ft, and for the larger triangle is x (AB). Wait, that would mean 280/140 = x/113, so x = 2*113 = 226. Ah, that makes sense! Because the two triangles are similar, with hypotenuses 280 and 140 (ratio 2:1), so the corresponding legs (on Oak) should be in ratio 2:1. The smaller triangle's leg (on Oak) is 113 ft? Wait, no, maybe the smaller triangle's leg (the one adjacent to the right angle) is 113 ft, and the larger one's is x (AB). So since the hypotenuses are in ratio 2:1, the legs should be in ratio 2:1. So x = 2*113 = 226? Wait, but 113*2 is 226, which is one of the options. So that must be it.

Step2: Calculate AB

Using the similarity of triangles, the ratio of the hypotenuses is \( \frac{280}{140} = 2 \). Therefore, the corresponding leg (AB) should be twice the corresponding leg of the smaller triangle (113 ft). So \( AB = 2 \times 113 = 226 \) ft.

Answer:

226 ft (corresponding to the option "226 ft")