To determine the location of the fruit market given that it is [tex]\(\frac{1}{4}\)[/tex] the distance from Taa's home to Lei's home, we will use the concept of the section formula. This formula helps find a point that divides a line segment between two points in a given ratio. Here’s the step-by-step solution:
1. Identify the coordinates of Taa’s and Lei’s homes:
- Taa’s home: [tex]\((x1, y1) = (4, 2)\)[/tex]
- Lei’s home: [tex]\((x2, y2) = (8, 8)\)[/tex]
2. Determine the ratio [tex]\( m:n \)[/tex]:
- The fruit market is [tex]\( \frac{1}{4} \)[/tex] of the distance from Taa's home to Lei's home, which can be expressed as [tex]\(1:3\)[/tex] in terms of section formula ratio, since [tex]\(1 + 3 = 4\)[/tex].
3. Apply the section formula to find the coordinates:
- The section formula for a point [tex]\(P(x, y)\)[/tex] dividing the line segment joining [tex]\((x1, y1)\)[/tex] and [tex]\((x2, y2)\)[/tex] in the ratio [tex]\(m:n\)[/tex] is:
[tex]\[
x = \frac{{m x2 + n x1}}{{m + n}}
\][/tex]
[tex]\[
y = \frac{{m y2 + n y1}}{{m + n}}
\][/tex]
4. Substitute the values into the formula:
- For [tex]\(x\)[/tex]-coordinate:
[tex]\[
x = \frac{{1 \cdot 8 + 3 \cdot 4}}{{1 + 3}} = \frac{{8 + 12}}{{4}} = \frac{20}{4} = 5
\][/tex]
- For [tex]\(y\)[/tex]-coordinate:
[tex]\[
y = \frac{{1 \cdot 8 + 3 \cdot 2}}{{1 + 3}} = \frac{{8 + 6}}{{4}} = \frac{14}{4} = 3.5
\][/tex]
5. Determine the nearest street and avenue:
- The [tex]\(x\)[/tex]-coordinate for the fruit market is 5, which corresponds to 5th Street.
- The [tex]\(y\)[/tex]-coordinate for the fruit market is 3.5, which we round to the nearest whole number, resulting in 4, corresponding to 4th Avenue.
Conclusion:
The fruit market is located at 5th Street and 4th Avenue.
