Question

solving a real-world problem
the 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?
○ 75 ft
○ 226 ft
○ 307 ft
○ 347 ft
first ave.
main
280 ft
second ave.
140 ft
b
113 ft
oak
a

Explanation:

Step1: Identify Similar Triangles

Since the avenues are parallel, the triangles formed are similar by the Basic Proportionality Theorem (Thales' theorem). Let the distance from A to B be \( x \). The ratios of corresponding sides of similar triangles are equal. So, \(\frac{280}{140}=\frac{x + 113}{113}\) (Wait, no, correct proportion: The larger triangle has a side of 280 ft, the smaller one 140 ft. The base of the larger triangle (along Oak) is \( AB + 113 \), and the smaller is 113? Wait, no, re-examine. Let's define: Let the distance from A to B be \( d \). The two triangles: one with hypotenuse 280, base \( d + 113 \); the other with hypotenuse 140, base 113. Wait, no, actually, the ratio of the hypotenuses is \( \frac{280}{140}=2 \). So the larger triangle is twice the smaller. Wait, no, maybe the vertical sides? Wait, no, the key is similar triangles: the ratio of the segments on Main (280 and 140) is 2:1, so the ratio of the segments on Oak should also be 2:1. Wait, maybe the correct proportion is \(\frac{280}{140}=\frac{d}{113}\)? No, that doesn't make sense. Wait, let's think again. The two triangles: one between First Ave, Main, and Oak; the other between Second Ave, Main, and Oak. Since First Ave and Second Ave are parallel (avenues run parallel), the triangles are similar. So the ratio of the sides of the larger triangle to the smaller is \( \frac{280}{140} = 2 \). So the base of the larger triangle (along Oak) is \( AB + 113 \), and the base of the smaller is 113? No, wait, maybe A is at the foot of First Ave, B at the foot of Second Ave. Then the distance from A to B is \( d \), and the two triangles: the larger triangle has hypotenuse 280, base \( d + 113 \), height (vertical) same? Wait, no, maybe the horizontal segments: the segment from the intersection of Main and Second Ave to B is 113, and from A to B is \( d \), and the hypotenuse of the smaller triangle is 140, larger is 280. So by similar triangles, \(\frac{280}{140}=\frac{d + 113}{113}\)? Wait, no, that would give \( 2=\frac{d + 113}{113} \), so \( d + 113 = 226 \), \( d = 113 \), which is not an option. Wait, maybe I got the proportion wrong. Let's use the Law of Similar Triangles correctly. Let’s denote:

Let the triangle with hypotenuse 280 ft have a base (along Oak) of length \( x \), and the triangle with hypotenuse 140 ft have a base of 113 ft. Since the triangles are similar (because avenues are parallel, so corresponding angles are equal), the ratio of hypotenuses is \( \frac{280}{140} = 2 \), so the ratio of bases should also be 2. Therefore, \( x = 2 \times 113 = 226 \)? Wait, no, that would mean the base of the larger triangle is 226, but that's the distance from A to B? Wait, maybe the distance from A to B is the base of the larger triangle, and the smaller triangle's base is 113, and hypotenuse 140, larger hypotenuse 280. So ratio 2:1, so base of larger is \( 2 \times 113 = 226 \)? Wait, but the options include 226 ft. Let's check:

If the two triangles are similar, then \(\frac{\text{Hypotenuse of larger triangle}}{\text{Hypotenuse of smaller triangle}}=\frac{\text{Base of larger triangle}}{\text{Base of smaller triangle}}\)

So \(\frac{280}{140}=\frac{AB}{113}\)

Wait, \( \frac{280}{140} = 2 \), so \( 2 = \frac{AB}{113} \), so \( AB = 2 \times 113 = 226 \) ft. Yes, that makes sense. So the distance Amana walks is 226 ft.

Step2: Verify the Proportion

We use the similarity of triangles (due to parallel avenues) to set up the proportion between the hypotenuses and the bases. The ratio of the hypotenuses (280/140 = 2) equals the ratio of the bases (AB/113). Solving for AB gives AB = 2 * 113 = 226 ft.

Answer:

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