Question

the diagram shows the intersections of several straight roads. the avenues run parallel to each other. amara 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

Explanation:

Step1: Use the basic proportionality theorem (Thales' theorem)

Since the avenues are parallel, we can set up a proportion. Let the distance from \(A\) to \(B\) be \(x\). The proportion is \(\frac{280}{140}=\frac{x + 113}{113}\)

Step2: Cross - multiply

Cross - multiplying gives \(280\times113=140\times(x + 113)\)

Step3: Simplify the equation

First, \(280\times113 = 31640\) and \(140\times(x + 113)=140x+140\times113=140x + 15820\)

So, \(31640=140x + 15820\)

Step4: Solve for \(x\)

Subtract \(15820\) from both sides: \(31640-15820 = 140x\)

\(15820=140x\)

Then \(x=\frac{15820}{140}=113\) (This is wrong. Let's use another approach - similar triangles)

Since the avenues are parallel, the two triangles are similar.

Let the distance \(AB = d\)

We know that \(\frac{d + 113}{113}=\frac{280 + 140}{140}\) (corresponding sides of similar triangles)

\(\frac{d + 113}{113}=\frac{420}{140}=3\)

\(d+113 = 3\times113\)

\(d+113=339\)

\(d = 339 - 113=226\)

Answer:

226 ft