Question

ABCD is a rectangle in the clockwise direction the coordinates of a are( 1,3) and C are( 5,1 )what is bmd lie on the line Y = 2 X + c then the coordinates of d are

Answer 1

Answer:

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Answer 2

Given:

In the given rectangle ,the coordinates of A(1,3)

The coordinates of C(5,1)

To Find:

The coordinates of D.

Solution:

The length of the diagonal AC =$\sqrt{(5-1)^{2}+(3-1)^{2})}=\sqrt{16+4}=2\sqrt{5}$

The diagonals AC and BD meet at a point O.

The coordinates of point O=(3,2)

The slope of the given line y=2x+c is 2.

So,Tan$\alpha$= m=2.

In case of rectangle, both the diagonals are equal.

So,AC=BD

BD=$2\sqrt{5}$

To calculate  a point from the point (3,2) at a distance of$\sqrt{5}$ we apply the formula,

$\dfrac{x-x1}{cos\alpha }=\dfrac{y-y1}{sin\alpha }=r$

Since we know , tanα=2

so Sinα=$\dfrac{2}{\sqrt{5}}$

Cosα=$\dfrac{1}{\sqrt{5}}$

$\dfrac{x-3}{\dfrac{1}{\sqrt{5}}}=\dfrac{y-2}{\dfrac{2} {\sqrt{5}}}=\sqrt{5}$

$\dfrac{x-3}{\dfrac{1}{\sqrt{5}}}=\sqrt{5}$

x-3=±1

x=3±1

x=4,2

Similarly for y ,

y-2=±2

y=2±2

y=4,0

So the required coordinates of B and D are (4,4) and (4,0).