Question

use the diagram below to answer questions 34 - 35.

1. find the distance from the bank to the grocery store.
2. find the distance from the coffee shop to the grocery store, then back to the bank.

Explanation:

Step1: Apply the Pythagorean theorem for question 34

The triangle formed by the bank, movie theater, and grocery store is a right - triangle. Let the distance from the bank to the grocery store be \(x\). According to the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 3\) (distance from bank to movie theater), \(b=x\) (distance from bank to grocery store), and \(c\) is the hypotenuse of the right - triangle.
For the right - triangle with legs \(3\) and \(x\) and hypotenuse equal to the length of the other right - triangle's hypotenuse (by the property of similar right - triangles or we can directly use \(x=\sqrt{d_{1}^{2}-d_{2}^{2}}\) where \(d_{1}\) is the length of the line from coffee shop to gas station (which is equal to the line from bank to grocery store + gas station to bank in terms of the right - triangle structure, but more simply, using the Pythagorean theorem for the right - triangle with legs \(3\) and \(x\) and we know from the other right - triangle (gas station - coffee shop - bank) that the relation of sides. The two right - triangles are similar. The distance from bank to grocery store \(x=\sqrt{8^{2}-5^{2}}\) (because the two right - triangles are similar and the length from coffee shop to gas station is \(8\) (hypotenuse of one right - triangle) and gas station to bank is \(5\)).

$$x=\sqrt{64 - 25}=\sqrt{39}\approx 6.24$$

Step2: For question 35

First, we know from question 34 that the distance from coffee shop to grocery store is \(8\) miles (hypotenuse of the first right - triangle) and the distance from grocery store to bank is \(\sqrt{39}\approx6.24\) miles.
The total distance \(D=8 + \sqrt{39}\approx8+6.24 = 14.24\)

Answer:

1. The distance from the bank to the grocery store is \(\sqrt{39}\approx6.24\) miles.
2. The distance from the coffee shop to the grocery store, then back to the bank is \(8+\sqrt{39}\approx14.24\) miles.