Question

The line segment joining the points (3, -4) and (1,2) is trisected at the points P and Q. If the coordinates of P and Q are (p, -2) and (5/3, q) respectively, find the values of p and q.​

Answer 1

Coordinate Geometry

As it is given that, the points P(p, -2) and Q(5/3, q) trisect the line segment joining the points A(3, -4) and B(1,2).

The point P trisection of line segment AB. So, the point P divides AB in the ratio of 1 : 2. And, The point Q trisection of line segment AB. So, the point Q divides AB in the ratio of 2 : 1.

[Please check the attachment for a better understanding]

Now, using the section formula. The coordinates of the point P(x, y) which divides the line segment joining A(x₁, y₁) and B(x₂, y₂) internally in the ratio m : m. Then, the coordinates of P will be:

$\implies\boxed{{(x,y) = \bigg(\dfrac{m_{1}x_{2}+m_{2}x_{1}}{m_{1}+m_{2}}, \dfrac{m_{1}y_{2}+m_{2}y_{1}}{m_{1}+m_{2}}\bigg)}}$

By substituting the values of $x_1 = 3$, $x_2 = 1$, $y_1 = -4$, $y_2 = 2$, $m = 1$ and $n = 2$ in the section formula. So, coordinate of P are;

$\implies (p, - 2) = \bigg( \dfrac{1(1)+2(3)}{1 + 2}, \dfrac{1(2)+2( - 4)}{1 + 2}\bigg) \\ \\ \implies (p, - 2) = \bigg( \dfrac{1+6}{1 + 2}, \dfrac{2+ (- 8)}{1 + 2}\bigg) \\ \\ \implies (p, - 2) = \bigg( \dfrac{7}{3}, \dfrac{2 - 8)}{1 + 2}\bigg) \\ \\ \implies (p, - 2) = \bigg( \dfrac{7}{3}, \dfrac{ - 6}{3}\bigg) \\ \\ \implies (p, - 2) = \bigg( \dfrac{7}{3}, 2 \bigg) \\ \\ \implies\boxed{ p = \dfrac{7}{3}}$

Similarly, By substituting the values of $x_1 = 3$, $x_2 = 1$, $y_1 = -4$, $y_2 = 2$, $m = 2$ and $n = 1$ in the section formula. So, coordinate of Q are;

$\implies \bigg( \frac{5}{3} , q \bigg) = \bigg( \dfrac{2(1)+1(3)}{1 + 2}, \dfrac{2(2)+1( - 4)}{1 + 2}\bigg) \\ \\ \implies \bigg( \frac{5}{3} , q \bigg) = \bigg( \dfrac{2+3}{1 + 2}, \dfrac{4+( - 4)}{1 + 2}\bigg) \\ \\ \implies \bigg( \frac{5}{3} , q \bigg) = \bigg( \dfrac{5}{3}, \dfrac{4 - 4)}{1 + 2}\bigg) \\ \\ \implies \bigg( \frac{5}{3} , q \bigg) = \bigg( \dfrac{5}{3}, \dfrac{0}{3}\bigg) \\ \\ \implies \bigg( \frac{5}{3} , q \bigg) = \bigg( \dfrac{5}{3}, 0\bigg) \\ \\ \implies \boxed{q = 0}$

Hence, the values of p and q are 7/3 and 0 respectively.

Answer 2

Answer:

Step-by-step explanation:

Please check the attached file