Solve for the following items:?

Question

Solve for the following items:

a. The point on the x-axis that is equidistant (equal distance) from (10,4) and (6,2).

b. The distance between the points (4,4) and (20,4).

c. The distance between the points (1,4) and (19,5).

d. The midpoint between (12, 4) and (18, 6)

e. The midpoint between (-8, -6) and (25, 3)

Answer 1

a)
The point on the x-axis is in the form (x, 0).

So, find the distance between (x, 0) and each of the given points.

(x, 0) to (10, 4):
d = sqrt [(x - 10)^2 + (0 - 4)^2]
= sqrt [x^2 - 20x + 100 + 16]
= sqrt (x^2 - 20x + 116)

(x, 0) to (6, 2)
d = sqrt [(x - 6)^2 + (0 - 2)^2]
= sqrt [x^2 - 12x + 26 + 4]
= sqrt (x^2 - 12x + 30)

Now, these two distances must be equal. So...
sqrt (x^2 - 20x + 116) = sqrt (x^2 - 12x + 30)
square both sides
x^2 - 20x + 116 = x^2 - 12x + 30
-20x + 116 = -12x + 30
-8x + 116 = 30
-8x = -86
x = 86 / 8 = 43 / 4 = 10.75

So, the point on the x-axis equidistant from the two given points is (10.75, 0)

b & c)
I've shown you (above) how to use the distance formula, so you can solve these two on your own now.

d)
Midpoint is the average of each part:
average of x-value:
(12 + 18) / 2 = 30 / 2 = 15

average of y-value:
(4 + 6) / 2 = 10 / 2 = 5

So, midpoint is (15, 5)

e) see part d - same process.

Answer 2

use these formulas:

m=(x1+x2)/2 , (y1+y2)/2

d=the square root of ( x1-x2)^2 + (ya-y2)^2

Answer 3

a. (8,3)
b. 16
c 18.028
d. (15, 5)
e. ( 8.5, -1.5)