# If the vectors PQ=-3i+4j+4k and PR=5i-2j+4k are the sides of a $\vartriangle PQR$, then the length of the median through P is

A.$\sqrt {14} $

B.$\sqrt {15} $

C.$\sqrt {17} $

D.$\sqrt {18} $

E.$\sqrt {19} $

Last updated date: 18th Mar 2023

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Answer

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Hint: We are going to solve the given problem using the median formula in the triangle. Median through any vertex divides the opposite side into two equal parts.

Given, PQ=-3i+4j+4k and PR=5i-2j+4k in triangle PQR

The median through vertex P divides QR side into two equal parts.

Let D is the midpoint of QR, then PD will be the length of median through vertex P.

$ \Rightarrow \overrightarrow {PQ} + \overrightarrow {PR} = 2\overrightarrow {PD} $

Therefore, the length of the median through P is

PD$ = \frac{1}{2}\left| {PQ + PR} \right|$

PD$ = \frac{1}{2}\left| {2i + 2j + 8k} \right|$

PD$ = \frac{1}{2}\sqrt {4 + 4 + 64} = \frac{1}{2}\sqrt {72} = \sqrt {18} $

$\therefore $ The length of the median through P = $\sqrt {18} $

Note: Median of a triangle is a line segment joining a vertex to the midpoint of the opposite side thus bisecting that side. In a triangle while using vectors the sum of two sides is equal to one particular side. $ \Rightarrow \overrightarrow {PQ} + \overrightarrow {PR} = \overrightarrow {QR} $, If D is the midpoint of the side QR, then QR = 2QD = 2RD. After drawing a line from P to D, the $\vartriangle PQR$ divided into two triangles both have PD side as common. Let’s take $\vartriangle PQD$$(or\vartriangle PRD)$ to find the length of PD.$ \Rightarrow PQ + QD = PD$, We know that QD is half of QR. Then we can find PD i.e., length of median through P.

Given, PQ=-3i+4j+4k and PR=5i-2j+4k in triangle PQR

The median through vertex P divides QR side into two equal parts.

Let D is the midpoint of QR, then PD will be the length of median through vertex P.

$ \Rightarrow \overrightarrow {PQ} + \overrightarrow {PR} = 2\overrightarrow {PD} $

Therefore, the length of the median through P is

PD$ = \frac{1}{2}\left| {PQ + PR} \right|$

PD$ = \frac{1}{2}\left| {2i + 2j + 8k} \right|$

PD$ = \frac{1}{2}\sqrt {4 + 4 + 64} = \frac{1}{2}\sqrt {72} = \sqrt {18} $

$\therefore $ The length of the median through P = $\sqrt {18} $

Note: Median of a triangle is a line segment joining a vertex to the midpoint of the opposite side thus bisecting that side. In a triangle while using vectors the sum of two sides is equal to one particular side. $ \Rightarrow \overrightarrow {PQ} + \overrightarrow {PR} = \overrightarrow {QR} $, If D is the midpoint of the side QR, then QR = 2QD = 2RD. After drawing a line from P to D, the $\vartriangle PQR$ divided into two triangles both have PD side as common. Let’s take $\vartriangle PQD$$(or\vartriangle PRD)$ to find the length of PD.$ \Rightarrow PQ + QD = PD$, We know that QD is half of QR. Then we can find PD i.e., length of median through P.

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