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

Question

If the components of \( \vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} \) along and perpendicular to \( \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) respectively are \( \frac{16}{11} (3\hat{i} + \hat{j} - \hat{k}) \) and \( \frac{1}{11} (-4\hat{i} - 5\hat{j} - 17\hat{k}) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to:

Answer 1

Given:

Vertices of ΔPQR are:
P (2, 1), Q (−2, 3), R (4, 5)

Step 1: Find the midpoint of PQ

Coordinates of midpoint M of PQ are given by:

M = ( (2 + (−2))/2 , (1 + 3)/2 )

M = (0 , 2)

Step 2: Find the slope of the median through R

Median through R joins R(4, 5) and M(0, 2)

Slope m =

m = (5 − 2) / (4 − 0) = 3/4

Step 3: Write the equation of the median

Using point–slope form:

y − 5 = (3/4)(x − 4)

Step 4: Simplify

4y − 20 = 3x − 12

3x − 4y + 8 = 0

Final Answer:

The equation of the median through vertex R is:
3x − 4y + 8 = 0

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

Solution and Explanation

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