Question

if point (p, q) is \\(\frac{1}{3}\\) of the way from a to b , what are the values of p and q? a(-3, -2), b(12, 4) grid image. p = \\(\square\\), q = \\(\square\\)

Explanation:

Step1: Recall the section formula

To find a point \((p, q)\) that divides the line segment joining \(A(x_1, y_1)\) and \(B(x_2, y_2)\) in the ratio \(m:n\), the coordinates are given by:
\[
p=\frac{mx_2 + nx_1}{m + n}, \quad q=\frac{my_2 + ny_1}{m + n}
\]
Here, the point is \(\frac{1}{3}\) of the way from \(A\) to \(B\), so the ratio \(m:n = 1:2\) (since the total parts are 3, with 1 part from \(A\) and 2 parts remaining to \(B\)). \(A(-3, -2)\) and \(B(12, 4)\), so \(x_1=-3\), \(y_1 = -2\), \(x_2 = 12\), \(y_2 = 4\), \(m = 1\), \(n = 2\).

Step2: Calculate \(p\)

Substitute the values into the formula for \(p\):
\[
p=\frac{1\times12+2\times(-3)}{1 + 2}=\frac{12-6}{3}=\frac{6}{3}=2
\]

Step3: Calculate \(q\)

Substitute the values into the formula for \(q\):
\[
q=\frac{1\times4+2\times(-2)}{1 + 2}=\frac{4-4}{3}=\frac{0}{3}=0
\]

Answer:

\(p = 2\), \(q = 0\)