Question 17 of 18intermediate🔧 ApplyNumerical3 marks

Find the coordinates of the points which divide the line segment joining A(-2, 2) and B(2, 8) into four equal parts.

Correct Answer

Let P, Q, and R be the points dividing the line segment joining A(-2, 2) and B(2, 8) into four equal parts. These points divide AB in the ratio 1:3, 1:1, and 3:1 respectively.

Coordinates of P (Ratio 1:3): Using the section formula, x=1(2)+3(−2)1+3=−44=−1x = \frac{1(2) + 3(-2)}{1 + 3} = \frac{-4}{4} = -1 and y=1(8)+3(2)1+3=144=3.5y = \frac{1(8) + 3(2)}{1 + 3} = \frac{14}{4} = 3.5. So, P is (−1,3.5)(-1, 3.5).

Coordinates of Q (Ratio 1:1): Using the section formula, x=1(2)+1(−2)1+1=0x = \frac{1(2) + 1(-2)}{1 + 1} = 0 and y=1(8)+1(2)1+1=5y = \frac{1(8) + 1(2)}{1 + 1} = 5. So, Q is (0,5)(0, 5).

Coordinates of R (Ratio 3:1): Using the section formula, x=3(2)+1(−2)3+1=1x = \frac{3(2) + 1(-2)}{3 + 1} = 1 and y=3(8)+1(2)3+1=6.5y = \frac{3(8) + 1(2)}{3 + 1} = 6.5. So, R is (1,6.5)(1, 6.5).

Therefore, the coordinates of the points are (−1,3.5)(-1, 3.5), (0,5)(0, 5), and (1,6.5)(1, 6.5).

Exercise: EXERCISE 7.2 | Q: 9 | (Page 13)
For More Understanding

Explanation

This problem applies the concept of dividing a line segment into equal parts, similar to the trisection method shown in Example 8 of the provided text. To divide a segment into four equal parts, we find points that divide the segment in the ratios 1:3, 1:1 (midpoint), and 3:1. The solution strictly uses the section formula provided in Example 6 (x=m1x2+m2x1m1+m2x = \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}) to calculate the coordinates for each ratio.

Solution Steps

  1. Step 1: Identify the ratios required to divide the segment into four equal parts (1:3, 1:1, and 3:1).

  2. Step 2: Apply the section formula for the ratio 1:3 to find the first point P.

  3. Step 3: Apply the section formula for the ratio 1:1 to find the midpoint Q.

  4. Step 4: Apply the section formula for the ratio 3:1 to find the last point R.

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