Question 19 of 70intermediate🔧 ApplyShort Answer3 marks

The vertices of ΔPQR\Delta PQR are P(2,1)P(2, 1), Q(−2,3)Q(-2, 3) and R(4,5)R(4, 5). Find equation of the median through the vertex RR.

Correct Answer

To find the equation of the median through vertex RR, we first determine the midpoint of the opposite side PQPQ. A median connects a vertex to the midpoint of the opposite side.

Step 1: Find the midpoint of PQ. Let MM be the midpoint of P(2,1)P(2, 1) and Q(−2,3)Q(-2, 3).

M=(2+(−2)2,1+32)=(0,2)M = \left(\frac{2 + (-2)}{2}, \frac{1 + 3}{2}\right) = (0, 2)

Step 2: Find the slope of the line through R and M. The median passes through R(4,5)R(4, 5) and M(0,2)M(0, 2). The slope mm is:

m=2−50−4=−3−4=34m = \frac{2 - 5}{0 - 4} = \frac{-3}{-4} = \frac{3}{4}

Step 3: Formulate the equation. Using the point-slope form y−y1=m(x−x1)y - y_1 = m(x - x_1) passing through R(4,5)R(4, 5):

y−5=34(x−4)y - 5 = \frac{3}{4}(x - 4) 4(y−5)=3(x−4)4(y - 5) = 3(x - 4) 4y−20=3x−124y - 20 = 3x - 12 3x−4y+8=03x - 4y + 8 = 0

Thus, the equation of the median is 3x−4y+8=03x - 4y + 8 = 0.

Exercise: EXERCISE 9.2 | Q: 8 | (Page 14)
For More Understanding

Explanation

The problem asks for the equation of a median. The provided context includes Question 5, which references finding the 'mid-point of the line segment', and Question 7, which involves finding the equation of a line 'Passing through the points'. These concepts are applied here. First, the midpoint of side PQ is calculated to determine the second point required for the median. Then, the slope is calculated using the vertex R and this midpoint. Finally, the equation of the line is derived using the slope and a point on the line.

Solution Steps

  1. Step 1: Identify the coordinates of vertices P(2, 1), Q(-2, 3), and R(4, 5).

  2. Step 2: Calculate the midpoint M of side PQ using the midpoint formula: M = ((2-2)/2, (1+3)/2) = (0, 2).

  3. Step 3: Calculate the slope (m) of the median passing through R(4, 5) and M(0, 2): m = (2-5)/(0-4) = 3/4.

  4. Step 4: Use the point-slope form y - 5 = (3/4)(x - 4) to find the equation.

  5. Step 5: Simplify the equation to standard form: 3x - 4y + 8 = 0.

PreviousNext