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Q9:
Find the coordinates of the points which divide the line segment joining A(– 2, 2) and B(2, 8) into four equal parts.
Solution :
Given: The coordinates of the endpoints of the line segment are $A(-2, 2)$ and $B(2, 8)$.
To Find: The coordinates of the three points $P$, $Q$, and $R$ that divide the line segment $AB$ into four equal parts.
Theoretical Basis: The Section Formula states that the coordinates of a point $P(x, y)$ dividing the line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $m:n$ are given by:
$x = \frac{mx_2 + nx_1}{m+n}$, $y = \frac{my_2 + ny_1}{m+n}$
Step 1: Finding point Q (The Midpoint of AB)
Since $Q$ divides $AB$ into four equal parts, $Q$ is the midpoint of $AB$. The ratio is $1:1$.
$x_Q = \frac{-2 + 2}{2} = \frac{0}{2} = 0$
$y_Q = \frac{2 + 8}{2} = \frac{10}{2} = 5$
Thus, $Q = (0, 5)$.
Step 2: Finding point P (The Midpoint of AQ)
Point $P$ is the midpoint of the segment $AQ$, where $A = (-2, 2)$ and $Q = (0, 5)$.
$x_P = \frac{-2 + 0}{2} = \frac{-2}{2} = -1$
$y_P = \frac{2 + 5}{2} = \frac{7}{2} = 3.5$
Thus, $P = (-1, 3.5)$.
Step 3: Finding point R (The Midpoint of QB)
Point $R$ is the midpoint of the segment $QB$, where $Q = (0, 5)$ and $B = (2, 8)$.
$x_R = \frac{0 + 2}{2} = \frac{2}{2} = 1$
$y_R = \frac{5 + 8}{2} = \frac{13}{2} = 6.5$
Thus, $R = (1, 6.5)$.
Conclusion: The points dividing the line segment $AB$ into four equal parts are $P(-1, 3.5)$, $Q(0, 5)$, and $R(1, 6.5)$.
Final Answer: The coordinates of the points are $(-1, 3.5)$, $(0, 5)$, and $(1, 6.5)$.
More Questions from Class 10 Mathematics Coordinate geometry EXERCISE 7.2
- Q1: Find the coordinates of the point which divides the join of (–1, 7) and (4, –3) in the ratio 2 : 3.
- Q10: Find the area of a rhombus if its vertices are (3, 0), (4, 5), (– 1, 4) and (– 2, – 1) taken in order. [Hint : Area of a rhombus = $\frac{1}{2}$ (product of its diagonals)]
- Q2: Find the coordinates of the points of trisection of the line segment joining (4, –1) and (–2, –3).
- Q3: To conduct Sports Day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1m each. 100 flower pots have been placed at a distance of 1m from each other along AD, as shown in Fig. 7.12. Niharika runs $\frac{1}{4}$th the distance AD on the 2nd line and posts a green flag. Preet runs $\frac{1}{5}$th the distance AD on the eighth line and posts a red flag. What is the distance between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag?
- Q4: Find the ratio in which the line segment joining the points (– 3, 10) and (6, – 8) is divided by (– 1, 6).
- Q5: Find the ratio in which the line segment joining A(1, – 5) and B(– 4, 5) is divided by the x-axis. Also find the coordinates of the point of division.
- Q6: If (1, 2), (4, $y$), ($x$, 6) and (3, 5) are the vertices of a parallelogram taken in order, find $x$ and $y$.
- Q7: Find the coordinates of a point A, where AB is the diameter of a circle whose centre is (2, – 3) and B is (1, 4).
- Q8: If A and B are (– 2, – 2) and (2, – 4), respectively, find the coordinates of P such that AP = $\frac{3}{7}$ AB and P lies on the line segment AB.
CBSE Solutions for Class 10 Mathematics Coordinate geometry
Chapters in CBSE - Class 10 Mathematics
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