Sure, let's break this down step by step to find the coordinates of the fruit market.
1. Identify the coordinates of Tia's and Lei's homes:
- Tia's home is at the corner of 4th Street and 8th Avenue. Hence, the coordinates are [tex]\((x_1, y_1) = (4, 8)\)[/tex].
- Lei's home is at the corner of 12th Street and 20th Avenue. Hence, the coordinates are [tex]\((x_2, y_2) = (12, 20)\)[/tex].
2. Understand the distance ratio:
- The fruit market is [tex]\(\frac{3}{4}\)[/tex] the distance from Tia's home to Lei's home. This means that the distance can be divided into 4 parts, where the fruit market is 3 parts away from Tia's home.
3. Apply the section formula to find the coordinates:
- The section formula is given by:
[tex]\[
x = \left(\frac{m}{m+n}\right) \left(x_2 - x_1\right) + x_1
\][/tex]
[tex]\[
y = \left(\frac{m}{m+n}\right) \left(y_2 - y_1\right) + y_1
\][/tex]
Where [tex]\(m = 3\)[/tex] and [tex]\(n = 1\)[/tex], since the distance ratio is [tex]\(3:1\)[/tex].
4. Calculate the x-coordinate of the fruit market:
[tex]\[
x = \left(\frac{3}{3+1}\right) (12 - 4) + 4
\][/tex]
[tex]\[
x = \left(\frac{3}{4}\right) \cdot 8 + 4
\][/tex]
[tex]\[
x = 6 + 4
\][/tex]
[tex]\[
x = 10
\][/tex]
5. Calculate the y-coordinate of the fruit market:
[tex]\[
y = \left(\frac{3}{3+1}\right) (20 - 8) + 8
\][/tex]
[tex]\[
y = \left(\frac{3}{4}\right) \cdot 12 + 8
\][/tex]
[tex]\[
y = 9 + 8
\][/tex]
[tex]\[
y = 17
\][/tex]
6. Interpret the results:
- The coordinates of the fruit market are [tex]\((10, 17)\)[/tex], meaning it is located at the corner of 10th Street and 17th Avenue.
Hence, the correct answer is:
10th Street and 17th Avenue.
