Question

If(x,3),(6,y),(8,2)and(9,4)are the vertices of parallelogram taken in order. then find the value of x and ythe centre of a circle is (-4,2).if one end of the diameter of the circle is (-3,7) then find the other end .

Answer 1

Answer:

The midpoint of BD is (152,y+42) ( 15 2 , y + 4 2 ) . Therefore, the value of x=7 and y=4 .

Step-by-step explanation:

Step 1: Consider, the given vertices of a parallelogram A B C D are:

$A=(x, 3) \quad B=(6, y) \quad C=(8,2) \quad D=(9,4)$

Since the diagonals of a parallelogram bisect each other. So, the midpoint of A C and B D will be the same.

Step 2: Find the midpoint of A C

$\begin{aligned}\text { Midpoint } & =\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) \\A C & =\left(\frac{x+8}{2}, \frac{3+2}{2}\right)\end{aligned}$

Thus, the midpoint of A C is $\left(\frac{x+8}{2}, \frac{3+2}{2}\right)$.

Step 3: Find the midpoint of BD

$\begin{aligned}\text { Midpoint } & =\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) \\B D & =\left(\frac{6+9}{2}, \frac{y+4}{2}\right)\end{aligned}$

Thus, the midpoint of B D is $\left(\frac{15}{2}, \frac{y+4}{2}\right)$.

Equate A C=B D to get x and y value

Step 4: Solve for x

$\begin{aligned}A C & =B D \\\left(\frac{x+8}{2}, \frac{5}{2}\right) & =\left(\frac{15}{2}, \frac{y+4}{2}\right) \\\frac{x+8}{2} & =\frac{15}{2} \\2(x+8) & =2 \cdot 15 \\2 x+16 & =30 \\2 x & =30-16 \\2 x & =14 \\x & =\frac{14}{2} \\\therefore x & =7\end{aligned}$

Step 5: Solve for y

$\begin{aligned}\frac{5}{2} & =\frac{y+4}{2} \\5 \cdot 2 & =2 \cdot(y+4) \\10 & =2 y+8 \\2 y & =10-2 \\2 y & =8 \\y & =\frac{8}{2} \\\therefore y & =4\end{aligned}$

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Answer 2

Answer:

(i) The value of x & y in parallelogram vertices are x = 7 & y = 1.

(ii) The coordinate of another end diameter of the circle is (-5,-3).

Step-by-step explanation:

(i) Parallelogram

Given, the vertices of the parallelogram are (x,3), (6,y), (8,2), and (9,4).

Let's find the value of x and y

The parallelogram's diagonals bisect each other, so

$(\frac{x + 8}{2}, \frac{3 + 2}{2}) = (\frac{9 + 6}{2}, \frac{4 + y}{2} )\\\\(\frac{x + 8}{2}, \frac{5}{2}) = (\frac{15}{2}, \frac{4 + y}{2} )$

Comparing both side coordinates, we get

$\frac{x + 8}{2} = \frac{15}{2}\\\\ x+8 = 15\\ x = 7\\and,\\\\\frac{4+y}{2} = \frac{5}{2}\\\\ 4+y = 5\\ y = 1$

Hence, the value of x & y is 7 & 1 respectively.

(ii) Circle

Given the center of the circle is (-4,2) and one coordinate of diameter is (-3,7)

Now, we have to find another coordinate of the diameter be (x,y)

Using the midpoint formula, we get

$\frac{x - 3}{2} = -4\\x-3 = -8\\x= -5\\\\and,\\\\\frac{y+7}{2} = 2\\ y+7 = 4\\y = -3$

Hence, the other coordinate of diameter is (-5,-3).

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