Question 42 of 70intermediate🔧 ApplyShort Answer4 marks

Find the equation of the right bisector of the line segment joining the points (3,4)(3, 4) and (−1,2)(-1, 2).

Correct Answer

The equation of the right bisector is 2x + y - 5 = 0.

Exercise: EXERCISE 9.3 | Q: 12 | (Page 18)
For More Understanding

Explanation

This question tests the concept of perpendicular bisector of a line segment. The right bisector passes through the midpoint of the segment and is perpendicular to it. The context (Example 13) establishes that a perpendicular bisector divides a line segment into two equal parts. Using the midpoint formula and the condition for perpendicular lines (product of slopes = -1), we derive the required equation.

Solution Steps

  1. Step 1: Find the midpoint of the line segment joining (3,4)(3, 4) and (−1,2)(-1, 2). Midpoint = (3+(−1)2,4+22)=(1,3)\left(\frac{3 + (-1)}{2}, \frac{4 + 2}{2}\right) = (1, 3)

  2. Step 2: Find the slope of the line segment. Slope m1=2−4−1−3=−2−4=12m_1 = \frac{2 - 4}{-1 - 3} = \frac{-2}{-4} = \frac{1}{2}

  3. Step 3: Find the slope of the right bisector (perpendicular to the segment). Slope m2=−1m1=−11/2=−2m_2 = -\frac{1}{m_1} = -\frac{1}{1/2} = -2

  4. Step 4: Write the equation using point-slope form. The right bisector passes through midpoint (1,3)(1, 3) with slope −2-2.

    y−3=−2(x−1)y - 3 = -2(x - 1) y−3=−2x+2y - 3 = -2x + 2 2x+y−5=02x + y - 5 = 0
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