+1 vote

# Suppose that P is the endpoint of a segment PQ and M is the midpoint of PQ. Find the coordinates of endpoint Q?

P(7,-4),M(8,5)
&
P(5.64,8.21),M(-4.04,1.60)

asked Jan 17, 2013 in GEOMETRY

+1 vote

1).  P(7,-4),M(8,5)

Let Q = (x, y)

Here P and Q are end points & M is mid point

P and Q mid point is M

M = P and Q mid point

So, mid point formula : [ (x₁+x₂)/2, (y₁+y₂)/2]

M (8, 5) ,  P (7, -4) and Q (x, y)

(8, 5) = [ (7 + x) / 2 ,  (-4 + y) / 2 ]

Then

8 = (7 + x) / 2 and 5 = (-4 + y) / 2

8 = (7 + x) / 2

Multiply each side by 2.

8 (2 ) = [(7 + x) / 2] (2)

Simplify

16 = 7 + x

Subtract 7 from each side.

16 - 7 = 7 + x - 7

9 = x

And

5 = (-4 + y) / 2

Multiply each side by 2.

5 (2) = [ (-4 + y) / 2](2)

10 = - 4 + y

10 + 4 = - 4 + y + 4

14 = y

There fore

Q(x, y) = (9, 14)

2). P(5.64,8.21),M(-4.04,1.60)

P(5.64, 8.21) ____________M(-4.04, 1.60)________________Q(x, y)

Let Q = (x, y)

Here P and Q are end points & M is mid point

P and Q mid point is M

M = P and Q mid point

So, mid point formula : [ (x₁+x₂)/2, (y₁+y₂)/2]

M (-4.04, 1.60) ,  P (5.64, 8.21) and Q (x, y)

(-4.04, 1.60) = [ (5.64 + x) / 2 ,  (8.21 + y) / 2 ]

Then

-4.04 = (5.64 + x) / 2 and 1.60 = (8.21 + y) / 2

-4.04 = (5.64 + x) / 2

Multiply each side by 2.

(-4.04) (2 ) = [(5.64 + x) / 2] (2)

Simplify

- 8.08 = 5.64 + x

Subtract 5.64 from each side.

- 8.08 - 5.64 = 5.64 + x - 5.64

- 13. 72 = x

And

1.60 = (8.21 + y) / 2

Multiply each side by 2.

(1.60) (2) = [ (8.21 + y) / 2](2)

3.2 = 8.21 + y

Subtract 8.21 from each side.

3.2 - 8.21 = 8.21 + y - 8.21

- 5.01 = y

There fore

Q(x, y) = (- 13. 72, -5.01)