Question

answer...............​

Answer 1

Gɪᴠᴇɴ :-

• Coordinates of A & B = (-14,-10) & (6,-2)
• P , Q & R divides Line segment AB in 4 Equal Parts.

Tᴏ Fɪɴᴅ :-

• Coordinates or P , Q & R ?

Fᴏʀᴍᴜʟᴀ ᴜsᴇᴅ :-

• Coordinates of Mid - Points of (x1,x2) & (y1,y2) is :- (x1+x2)/2 & (y1+y2)/2

Sᴏʟᴜᴛɪᴏɴ :-

A(-14,10)--------P(a,b)--------Q(c,d)-------R(e,f)---------B(6,-2)

As we can see Point Q is in The Middle of AB .

So,

Coordinates of Q :-

→ c = [(-14) + 6]/2 = (-8)/2 = (-4)

→ d = [(-10) + (-2)]/2 = (-12)/2 = (-6) .

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Now we have :-

A(-14,-10)--------P(a,b)--------Q(-4,-6)

Here, As we can see P is in The Middle of AQ,

So, Coordinates of P :-

→ a = [(-14) + (-4)]/2 = (-18)/2 = (-9)

→ b = [-10 + (-6)]/2 = (-16)/2 = (-8) .

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Now we have :-

Q(-4,-6)--------R(e,f)--------B(6,-2)

Here, As we can see R is in The Middle of QB,

So, Coordinates of R :-

→ e = [(-4) + 6]/2 = 2/2 = 1.

→ f = [(-6) + (-2)]/2 = (-8)/2 = (-4).

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Hence, we can conclude That, The Required Coordinates which Divide Line Segment AB in 4 Equal Parts are :- P(-9,-8) , Q(-4,-6) & R(1,-4) . (Ans.)

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Answer 2

${\huge{\bf{\red{\underline{Solution:}}}}}$

${\bf{\blue{\underline{Given:}}}}$

• Line segment A(-14,-10),B(6,-2)

${\bf{\blue{\underline{To\:Find:}}}}$

• Coordinates of P(x,y),Q(x,y),R(x,y)

${\bf{\blue{\underline{Now:}}}}$

• Let the P,Q,R be thr points on line segment AB
• Therefore AP = PQ = QR = RB.
• P divides line AB in the ratio 1:3

$\bigstar \boxed{\sf{ \purple{P(x,y) = \bigg(\frac{m_{1}x_{2} + m_{2}x_{1}}{m_{1} + m_{1}} , \frac{m_{1}y_{2} + m_{2}y_{1}}{m_{1} + m_{2}} \bigg)}}}$

$: \implies{\sf{ \bigg( \frac{1(6) + 3( - 14)}{1 + 3} } ,\frac{1( - 2) + 3 ( - 10)}{1 + 3} \bigg) }$

$: \implies{\sf{ \bigg( \frac{6 - 42}{4} , \frac{ - 2 - 30}{4} \bigg)}}$

$: \implies{\sf{ \bigg( \frac{ - 36}{4} , \frac{ - 32}{4} \bigg) }}$

$: \implies{\sf{ P(x,y)=\big( - 9, - 8 \big) }}$

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• Q divides line AB in the ratio 2:2 Q(x,y).

$\star{\sf{ Q(x,y) = \: \bigg( \frac{2(6) + 2( -14)}{2 + 2} , \frac{2( - 2) + 2( - 10)}{2 + 2} \bigg)}}$

$: \implies{\sf{ \bigg(\frac{12 - 28}{4}, \frac{ - 4 - 20}{4} \bigg) }}$

$: \implies{\sf{ \bigg(\frac{ - 16}{4},\frac{ - 24}{4} \bigg) }}$

$: \implies{\sf{ Q(x,y)= \big( - 4, - 6 \big) }}$

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• P divides line AB in the ratio 3:1.

$: \implies{\sf{ R(x,y)=\bigg( \frac{3(6) + 1( - 14)}{3 + 1} } .\frac{3( - 2) + 1 ( - 10)}{ 3 + 1} \bigg) }$

$: \implies{\sf{ \bigg( \frac{18 - 14}{4} , \frac{ - 6 - 10}{4} \bigg)}}$

$: \implies{\sf{ \bigg( \frac{4}{4} , \frac{ - 16}{4} \bigg)}}$

$: \implies{\sf{R(x,y) =\big( 1, - 4 \big) }}$

Hence the coordinates of P(x,y)= (-9,-8) andQ(x,y)= (-4-6) and R(x,y)=(1,-4)