Alright! Let's solve this step-by-step.
First, we identify the coordinates of Tia's and Lei's homes:
- Tia lives at the corner of 4th Street and 8th Avenue, so her home is at [tex]\((x_1, y_1) = (4, 8)\)[/tex].
- Lei lives at the corner of 12th Street and 20th Avenue, so his home is at [tex]\((x_2, y_2) = (12, 20)\)[/tex].
The fruit market is [tex]\(\frac{3}{4}\)[/tex] the distance from Tia's home to Lei's home.
We are given the formulas to find the coordinates [tex]\((x, y)\)[/tex] where the fruit market is located:
[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]
Here, [tex]\(m = 3\)[/tex] and [tex]\(n = 1\)[/tex].
Let's find the [tex]\(x\)[/tex] coordinate first:
[tex]\[
(x_2 - x_1) = 12 - 4 = 8
\][/tex]
[tex]\[
\left( \frac{m}{m+n} \right) = \left( \frac{3}{3+1} \right) = \left( \frac{3}{4} \right)
\][/tex]
[tex]\[
x= \left( \frac{3}{4} \right) \left(8\right) + 4 = 6 + 4 = 10
\][/tex]
So, the [tex]\(x\)[/tex] coordinate is [tex]\(10\)[/tex].
Now, let's find the [tex]\(y\)[/tex] coordinate:
[tex]\[
(y_2 - y_1) = 20 - 8 = 12
\][/tex]
[tex]\[
\left( \frac{m}{m+n} \right) = \left( \frac{3}{3+1} \right) = \left( \frac{3}{4} \right)
\][/tex]
[tex]\[
y= \left( \frac{3}{4} \right) \left(12\right) + 8 = 9 + 8 = 17
\][/tex]
So, the [tex]\(y\)[/tex] coordinate is [tex]\(17\)[/tex].
Thus, the coordinates of the fruit market are [tex]\((10, 17)\)[/tex], meaning it is located at the corner of 10th Street and 17th Avenue.
Therefore, the correct answer is:
10th Street and 17th Avenue
