Question

point a has coordinates (-5, 3). if point (1, 6) is \\(\frac{3}{4}\\) of the way from a to b, what are the coordinates of point b?
(\square, \square)

Explanation:

Step1: Recall the section formula

The section formula for a point \( P(x, y) \) that divides the line segment joining \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \) is given by:
\[
x=\frac{mx_2 + nx_1}{m + n}, \quad y=\frac{my_2 + ny_1}{m + n}
\]
Here, the point \( (1, 6) \) is \( \frac{3}{4} \) of the way from \( A \) to \( B \), so the ratio \( m:n = 3:1 \) (since the remaining part is \( 1-\frac{3}{4}=\frac{1}{4} \), so the ratio of the part from \( A \) to the point and from the point to \( B \) is \( 3:1 \)). \( A(-5, 3) \), let \( B=(x_2, y_2) \), and the point \( P=(1, 6) \).

Step2: Solve for \( x_2 \)

Using the formula for \( x \)-coordinate:
\[
1=\frac{3x_2+1\times(-5)}{3 + 1}
\]
Multiply both sides by \( 4 \):
\[
4=3x_2-5
\]
Add \( 5 \) to both sides:
\[
3x_2=4 + 5=9
\]
Divide by \( 3 \):
\[
x_2 = 3
\]

Step3: Solve for \( y_2 \)

Using the formula for \( y \)-coordinate:
\[
6=\frac{3y_2+1\times3}{3 + 1}
\]
Multiply both sides by \( 4 \):
\[
24=3y_2+3
\]
Subtract \( 3 \) from both sides:
\[
3y_2=24 - 3=21
\]
Divide by \( 3 \):
\[
y_2 = 7
\]

Answer:

\( (3, 7) \)