Question

if P (2,2) Q(-4,-4) and R (5,-8) are the vertices of a triangle PQR then what is the length of median through R ?

Answer 1

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✳ Required Answer:

✏ GiveN:

• Vertices of the triangle are P(2,2), Q(-4,-4) and R(5,-8)
• A median is drawn from R to the opposite side.

✏ To FinD:

• Length of the median from R....?

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✳ How to solve?

Before solving the question, let's know about median.

$\large{ \bold{ \underline{ \underline{ \red{Median}}}}}$

The line segment joining the midpoint of a side to the vertex opposite to the side is called a median.

Here, we have to find the midpoint of the side opposite to R, which can be found by Midpoint formula and then we will get our two Required points that are R and the midpoint of other side. Then, we will find the length of the median by using the distance formula.

So here are the required formulas,

$\large{ \bold{ \underline{ \underline{ \red{Midpoint \: formula}}}}} \\ \\ \rm{Midpoint \: of \: line \: joining \: (x_1 \:, y_1) \: and \: (x_2 \:, y_2)} \\ \\ \large{ \boxed{ \rm{midpoint = ( \frac{x_1 + x_2}{2} \:, \frac{y_1 + y_2}{2} )}}}$

And,

$\large{ \bold{ \underline{ \underline{ \red{Distance \: formula}}}}} \\ \\ \rm{Distance \: between \: points \: (x_1 \: y_1) \: and \: (x_2 \: y_2)} \\ \\ \large{ \boxed{ \rm{Distance = \sqrt{(x_2 - x_1) {}^{2} + (y_2 - y_1) {}^{2} } }}}$

So, By using these formula, we can solve the above question and find the length of the median from R.

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✳ Solution:

✏ Refer to the attachment....

$\large{\triangle}$PQR is our triangle. Let median passing through R to opposite side PQ meet at M. Then, here our median is RM.

Finding coordinates of M,

PM = QM

Given M is the midpoint of PQ and P(2,2) and Q(-4,-4)

By using midpoint formula,

$\large{ \rm{ \rightarrow \: M( \frac{x_1 + x_2}{2} \: , \frac{y_1 + y_2}{2} )}} \\ \\ \large{ \rm{ \rightarrow \: M( \frac{2 + - 4}{2} \: , \frac{2 + - 4}{2} )}} \\ \\ \large{ \rm{ \rightarrow \: M( \frac{ - 2}{2} \: , \frac{ - 2}{2} )}} \\ \\ \large{ \rm{ \rightarrow \: M( - 1 \:, - 1)}}$

Now, we have our Required points of endpoints of median RM i.e. R(5,-8) and M(-1,-1). So finding the length of median RM.

By using distance formula,

$\large{ \rm{ \rightarrow \: RM = \sqrt{( - 1 - 5) {}^{2} + ( - 1 - ( - 8)) {}^{2} } }} \\ \\ \large{ \rm{ \rightarrow \: RM = \sqrt{( - 6) {}^{2} + ( - 1 + 8) {}^{2} } }} \\ \\ \large{ \rm{ \rightarrow \: RM = \sqrt{( - 6) {}^{2} + (7) {}^{2} } \: cm}} \\ \\ \large{ \rm{ \rightarrow \: RM = \sqrt{36 + 49} \: cm}} \\ \\ \large{ \rm{ \rightarrow \: rm = \boxed{ \purple{ \rm{\sqrt{85} \: cm}}}}}$

✏Thus, our required length of median RM = $\large{\rm{\sqrt{85} \:cm}}$

$\large{ \therefore{ \underline{ \underline{ \green{ \rm{Hence \: solved \: \dag}}}}}}$

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Answer 2

Let RA be the median of ∆PQR through R.

Since we know that ,,,

Median bisect the base

i.e., PA = QA

i.e. A is the mid point of PQ

$Mid \: point \: PQ \: (a) =( \frac{x _{1}+ x _{2}}{2} , \frac{y _{1} + y _{2} }{2} )\\ \\ = ( \frac{2 + ( - 4)}{2} ,\frac{2 + ( - 4)}{2} ) \\ \\ =( \frac{ - 2}{2} , \frac{ - 2}{2} ) \\ \\ a= ( - 1,- 1)$

$Distance \: \: AR \: = \sqrt{{(x _{2} - x _{1}) }^{2} + ( {y _{2} - y _{1}) }^{2} } \\ \\ = \sqrt{ {( - 1 - 5)}^{2} +{ ( - 1 - ( - 8) }^{2} } \\ \\ = \sqrt{ { ( - 6)}^{2} + {(7)}^{2} } \\ \\ = \sqrt{36 + 49} \\ \\ AR= \sqrt{85}$

Therefore ,,

The length of median through R is √85 units