Certainly! Let's solve this step-by-step using the information provided.
1. Understand the Problem:
Krystal is crossing an intersection. We have:
- The length of the diagonal of this intersection is 12 meters.
- The width of one of the streets is 9 meters.
We need to find out the width of the second street.
To find the width of the second street, we can use the Pythagorean Theorem, which is applicable here since the diagonal forms a right triangle with the widths of the two streets.
2. Pythagorean Theorem:
The Pythagorean Theorem states:
[tex]\[
c^2 = a^2 + b^2
\][/tex]
Where:
- [tex]\(c\)[/tex] is the length of the hypotenuse (diagonal of the intersection),
- [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are the lengths of the legs of the right triangle (widths of the two streets).
In our case:
- [tex]\(c = 12\)[/tex] meters,
- [tex]\(a = 9\)[/tex] meters,
- [tex]\(b\)[/tex] is the width of the second street, which we need to find.
3. Set up the Equation:
We can rearrange the Pythagorean formula to solve for [tex]\(b\)[/tex]:
[tex]\[
b^2 = c^2 - a^2
\][/tex]
Substituting the known values:
[tex]\[
b^2 = 12^2 - 9^2
\][/tex]
4. Calculate:
Calculate the squares of the known values:
[tex]\[
12^2 = 144
\][/tex]
[tex]\[
9^2 = 81
\][/tex]
Subtract to find [tex]\(b^2\)[/tex]:
[tex]\[
b^2 = 144 - 81 = 63
\][/tex]
Take the square root to find [tex]\(b\)[/tex]:
[tex]\[
b = \sqrt{63}
\][/tex]
5. Simplify the Square Root:
Calculate [tex]\(\sqrt{63}\)[/tex]:
[tex]\[
\sqrt{63} \approx 7.94
\][/tex]
Therefore, the width of the second street is approximately 7.94 meters.
6. Visual Representation:
Here is a simplified drawing to help visualize the problem:
```
_____________________
| Diagonal 12m |
| \ |
| \ |
| \ |
9m | \ | 7.94m
| \ |
|______\_______________|
Street 1 Street 2
```
In this drawing, we have a right-angled triangle where:
- One leg is 9 meters (width of Street 1).
- The hypotenuse (diagonal) is 12 meters.
- The other leg (width of Street 2) is approximately 7.94 meters.
Thus, Krystal can deduce that the second street is about 7.94 meters wide.
