Question

Three vertices of a parallelogram taken in order are (-1, -6), (2,-5) and (7,2).The fourth vertex is
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Answer 1

♣ Qᴜᴇꜱᴛɪᴏɴ :

• Three vertices of a parallelogram taken in order are (-1, -6), (2,-5) and (7,2).The fourth vertex is

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♣ ᴀɴꜱᴡᴇʀ :

• The co-ordinates of fourth vertex is (4, 1)

$\setlength{\unitlength}{1 cm}\begin{picture}(20,15)\thicklines\qbezier(1,1)(1,1)(6,1)\qbezier(1,1)(1,1)(1.6,4)\qbezier(1.6,4)(1.6,4)(6.6,4)\qbezier(6,1)(6,1)(6.6,4)\qbezier(6.6,4)(6.6,4)(1,1)\qbezier(1.6,4)(1.6,4)(6,1)\put(0.7,0.5){\sf A (-1,-6)}\put(6,0.5){\sf B (2,-5)}\put(1.4,4.3){\sf D (4,1)}\put(6.6,4.3){\sf C (7,2)}\end{picture}$

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♣ ᴄᴀʟᴄᴜʟᴀᴛɪᴏɴꜱ :

The given points :

• A (-1 , -6)
• B (2, -5)
• C (7 ,2 )

$\setlength{\unitlength}{1 cm}\begin{picture}(20,15)\thicklines\qbezier(1,1)(1,1)(6,1)\qbezier(1,1)(1,1)(1.6,4)\qbezier(1.6,4)(1.6,4)(6.6,4)\qbezier(6,1)(6,1)(6.6,4)\qbezier(6.6,4)(6.6,4)(1,1)\qbezier(1.6,4)(1.6,4)(6,1)\put(0.7,0.5){\sf A (-1,-6)}\put(6,0.5){\sf B (2,-5)}\put(1.4,4.3){\sf D}\put(6.6,4.3){\sf C (7,2)}\end{picture}$

Now we have to find the co-ordinates of fourth vertex "D"

Fourth vertex of a parallelogram is given by :

$\sf{D(x_1-x_2+x_3,\:y_1-y_2+y_3)}$

Also, we have :

• x₁ = -1
• x₂ = 2
• x₃ = 7
• y₁ = -6
• y₂ = -5
• y₃ = 2

Substituting the values in $\sf{D(x_1-x_2+x_3,\:y_1-y_2+y_3)}$ :

$\sf{D(-1-2+7,\:-6-(-5)+2)}$

$\implies\sf{D(-3+7,\:-6+5+2)}$

$\implies\sf{D(4,\:1)}$

Hence the co-ordinates of fourth vertex is (4, 1)

Answer 2

Given:

The parallelogram taken in order are (-1,-6),(2,-5),(7,2).

To find:

The fourth vertex of parallelogram.

Step-by-step explanation:

Let A(-1,-6),B(2,-5),C(7,2)and D(x, y).

AC and BD are diagonals

so, that diagonals are divided in ratio 1:1

Mid-point of AC=Mid-point BD

$(\frac{-1+7}{2} ,\frac{-6+2}{2} )=(\frac{2+x}{2} \frac{-5+y}{2})$

$(3,-2)=(\frac{2+x}{2} ,\frac{-5+y}{2}$

Now, comparing co-ordinates

$3=\frac{2+x}{2}$

$6=2+x$

$x=4$

Now,$-2=\frac{-5+y}{2}$

$-4=-5+y$

$y=1$

therefore, The co-ordinates of fourth vertex of parallelogram D(4,1).