NCERT Class X Chapter 7: Coordinate Geometry Exercise 7.2 Question 9
NCERT Class X Chapter 7: Coordinate Geometry
Question:
Find the coordinates of the points which divide the line segment joining A(–2, 2) and B(2, 8) into four equal parts.
Given:
Points \( A(-2,\,2) \) and \( B(2,\,8) \).
To Find:
The coordinates of the points that divide the line segment \( AB \) into four equal parts.
Formula:
Section formula: The coordinates of the point \( P \) dividing the line segment joining \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \) are:
$$ P = \left( \frac{mx_2 + nx_1}{m+n}, \; \frac{my_2 + ny_1}{m+n} \right) $$Solution:
Step 1: Let the points dividing \( AB \) into four equal parts be \( P \), \( Q \), and \( R \). These points divide \( AB \) at 1/4, 1/2, and 3/4 of the way from \( A \) to \( B \).
Step 2: The required points divide \( AB \) in the ratios 1:3, 1:1, and 3:1 respectively. Let us calculate the coordinates for each.
Step 3: For the point \( P_1 \) dividing \( AB \) in the ratio \( 1:3 \):
$$ x = \frac{1 \times 2 + 3 \times (-2)}{1+3} = \frac{2 - 6}{4} = \frac{-4}{4} = -1 \\ y = \frac{1 \times 8 + 3 \times 2}{1+3} = \frac{8 + 6}{4} = \frac{14}{4} = 3.5 \\ \Rightarrow P_1(-1,\,3.5) $$Step 4: For the point \( P_2 \) dividing \( AB \) in the ratio \( 1:1 \):
$$ x = \frac{1 \times 2 + 1 \times (-2)}{1+1} = \frac{2 - 2}{2} = 0 \\ y = \frac{1 \times 8 + 1 \times 2}{1+1} = \frac{8 + 2}{2} = \frac{10}{2} = 5 \\ \Rightarrow P_2(0,\,5) $$Step 5: For the point \( P_3 \) dividing \( AB \) in the ratio \( 3:1 \):
$$ x = \frac{3 \times 2 + 1 \times (-2)}{3+1} = \frac{6 - 2}{4} = \frac{4}{4} = 1 \\ y = \frac{3 \times 8 + 1 \times 2}{3+1} = \frac{24 + 2}{4} = \frac{26}{4} = 6.5 \\ \Rightarrow P_3(1,\,6.5) $$Step 6: To divide into four equal parts, we need three points. But the question asks for the coordinates of the points which divide the segment into four equal parts. The points are at 1/4, 1/2, and 3/4 of the way from \( A \) to \( B \). Let's use the section formula with ratios 1:3, 1:1, and 3:1, but also check for the correct points.
Step 7: Alternatively, we can use the formula for internal division at \( \frac{k}{4} \) for \( k = 1,2,3 \):
$$ x = x_1 + \frac{k}{4}(x_2 - x_1) \\ y = y_1 + \frac{k}{4}(y_2 - y_1) $$ For \( k = 1 \): $$ x = -2 + \frac{1}{4}(2 - (-2)) = -2 + 1 = -1 \\ y = 2 + \frac{1}{4}(8 - 2) = 2 + 1.5 = 3.5 \\ (-1,\,3.5) $$ For \( k = 2 \): $$ x = -2 + \frac{2}{4}(2 - (-2)) = -2 + 2 = 0 \\ y = 2 + \frac{2}{4}(8 - 2) = 2 + 3 = 5 \\ (0,\,5) $$ For \( k = 3 \): $$ x = -2 + \frac{3}{4}(2 - (-2)) = -2 + 3 = 1 \\ y = 2 + \frac{3}{4}(8 - 2) = 2 + 4.5 = 6.5 \\ (1,\,6.5) $$Step 8: However, to divide the segment into four equal parts, we need the points at 1/4, 1/2, and 3/4 of the way. Alternatively, we can write the points as:
$$ \text{At } \frac{1}{4}: \; x = -2 + 1 = -1, \; y = 2 + 1.5 = 3.5 \\ \text{At } \frac{2}{4}: \; x = -2 + 2 = 0, \; y = 2 + 3 = 5 \\ \text{At } \frac{3}{4}: \; x = -2 + 3 = 1, \; y = 2 + 4.5 = 6.5 $$ So, the points are \( (-1, 3.5),\; (0, 5),\; (1, 6.5) \).Step 9: If we interpret "divide into four equal parts" as finding the three points that split the segment into four equal segments, the answer is \( (-1, 3.5),\; (0, 5),\; (1, 6.5) \).
Result:
The coordinates of the points which divide the line segment joining \( A(-2,\,2) \) and \( B(2,\,8) \) into four equal parts are:
\( (-1,\,3.5),\; (0,\,5),\; (1,\,6.5) \)
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