Let's analyze the problem step by step.
1. Identifying Coordinates:
- Tia's home coordinates: (4, 8)
- Lei's home coordinates: (12, 20)
2. Fractional Distance:
The fruit market is [tex]\(\frac{3}{4}\)[/tex] the distance from Tia's home to Lei's home. This can be understood as splitting the distance into a ratio of [tex]\( \frac{m}{m+n} = \frac{3}{4} \)[/tex]. Using this ratio:
- [tex]\(m = 3\)[/tex]
- [tex]\(n = 1\)[/tex] (since [tex]\(4 - 3 = 1\)[/tex])
3. Formula Application:
Using the given formulas,
[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]
4. Substitute the Known Values:
- For [tex]\(x\)[/tex]:
[tex]\[
x = \left(\frac{3}{3+1}\right) \left(12 - 4\right) + 4
\][/tex]
Simplifying inside the parentheses:
[tex]\[
x = \left(\frac{3}{4}\right) \left(8\right) + 4
\][/tex]
[tex]\[
x = 6 + 4 = 10
\][/tex]
- For [tex]\(y\)[/tex]:
[tex]\[
y = \left(\frac{3}{3+1}\right) \left(20 - 8\right) + 8
\][/tex]
Simplifying inside the parentheses:
[tex]\[
y = \left(\frac{3}{4}\right) \left(12\right) + 8
\][/tex]
[tex]\[
y = 9 + 8 = 17
\][/tex]
Therefore, the coordinates of the fruit market are (10, 17).
Among the given options:
- 10th Street and 17th Avenue
Thus, the fruit market is at 10th Street and 17th Avenue.
