## Find the coordinates of the points which divide the line segment joining

[question] Question. Find the coordinates of the points which divide the line segment joining A (– 2, 2) and B (2,8) into four equal parts. [/question] [solution] Solution: Here, the given points are A(–2, 2) and B(2, 8) Let $P_{1}, P_{2}$ and $P_{3}$ divide $A B$ in four equal parts. $\because \quad \mathrm{AP}_{1}=\mathrm{P}_{1} \mathrm{P}_{2}=\mathrm{P}_{2} \mathrm{P}_{3}=\mathrm{P}_{3} \mathrm{~B}$ Obviously, $\mathrm{P}_{2}$ is the mid-point of $\mathrm{AB}$ $\therefore \quad$ Coordinates o...

## Find the ratio in which the line segment joining the points

[question] Question. Find the ratio in which the line segment joining the points (– 3, 10) and (6, –8) is divided by (–1, 6). [/question] [solution] Solution: Let the required ratio be K : 1 [/solution]...

## Find the coordinates of the points of trisection of the line segment joining

[question] Question. Find the coordinates of the points of trisection of the line segment joining $(4,-1)$ and $(-2,-3)$. [/question] [solution] Solution: Points P and Q trisect the line segment joining the points A(4, – 1) and B(2, – 3), i.e., AP = PQ = QB. Here, P divides AB in the ratio 1 : 2 and Q divides AB in the ratio 2 : 1. $x$-coordinate of $P=\frac{1 \times(-2)+2 \times(4)}{1+2}=\frac{6}{3}=2$; $y-$ coordinate of $P=\frac{1 \times(-3)+2 \times(-1)}{1+2}=\frac{-5}{3}$ Thus, the coordina...

## Check whether (5, – 2), (6, 4) and (7, – 2) are the vertices of an isosceles triangle.

[question] Question. Check whether (5, – 2), (6, 4) and (7, – 2) are the vertices of an isosceles triangle. [/question] [solution] Solution: Let the points be A(5, –2), B(6, 4) and C(7, –2). $\therefore \quad \mathrm{AB}=\sqrt{(\mathbf{6}-\mathbf{5})^{2}+\mathbf{4}-(\mathbf{- 2})^{2}}$ $=\sqrt{(1)^{2}+(B)^{2}}=\sqrt{1+36}=\sqrt{37}$ $B C=\sqrt{(7-6)^{2}+(-2-4)^{2}}$ $=\sqrt{(1)^{2}+(-6)^{2}}=\sqrt{1+36}=\sqrt{37}$ $A C=\sqrt{(7-5)^{2}+(-2-(-2))^{2}}$ $=\sqrt{(+2)^{2}+(0)^{2}}=\sqrt{4+0}=2$ We ha...