Question

A circle has a diameter with endpoints (-8, 2) and (-2, 6).

What is the equation of the circle?


r2 = (x - 3)2 + (y + 4)2
r2 = (x - 5)2 + (y + 4)2
r2 = (x + 5)2 + (y - 4)2
r2 = (x + 3)2 + (y - 4)2

Answer 1

I would find the average of the x values of (-8, 2) and (-2, 6)
and the average of the y values of those two points. that will be the center of the circle
if you get (h,k)
the equation would be
(x-h)^2 + (y-k)^2 = r^2

Answer 2

What is (h.k).. I neveer quite understood that...

Answer 3

(h, k) is the center, but I'll let Phi take over :p

Answer 4

(h,k) is just to show a pair of numbers. we use (h,k) as the (x,y) values of the center point of the circle.

but before worrying about that, can you find the average of -8 and -2 ?

Answer 5

(to find the average , add up the two numbers, and then divide by 2)

Answer 6

yeah I have x=-5 and y= 4

Answer 7

now you use those numbers in the "template"
(x-h)^2 + (y-k)^2 = r^2

where (h,k) is your (-5,4)

Answer 8

(x+5)^2 + (y-4)^2=r^2

Answer 9

yes, and if we wanted we could find the radius using the distance formula between the center and one of the points they gave us. I think we get r^2 = 13

Answer 10

Ok thank you!! Could you help with some more please...

Answer 11

we could use for example
\[ ( x_0, y_0) \]
instead of (h,k) to show a pair of numbers
but it's harder to type

Answer 12

The vertices of an isosceles triangle are A(3, 6), B(7, 2), and C(4, 3).

What is the equation of the triangle's line of symmetry?


y = x + 1
y = x - 1
y = x
y = -x + 1

Answer 13

I would plot the 3 points first, to see what is going on.

Answer 14

can you sketch the plot ?

Answer 15

Yeah, hold on

Answer 16

Alright..now what?

Answer 17

the "axis of symmetry" is the line where you can fold the triangle in half, and the two sides lay on top of each other.

Answer 18

any idea which point this "axis" goes through ?

Answer 19

Point C is the middle point of the angle...

Answer 20

yes. any idea what angle this "axis" makes with the "base" side i.e line segment AB ?

Answer 21

An obtuse angle...?

Answer 22

the axis is the same as the line that bisects the base, and forms a right angle with the base.
It's easier to see if you rotate the triangle so it looks "normal"

any way, we can find the slope of the axis, because it is perpendicular to line AB
if line AB has slope m, the axis will have slope -1/m
(negative reciprocal)

I would find the slope of AB
flip it and multiply by -1
that will be the slope of the axis
then use

y = mx + b
put in the point (4,3) and solve for b

Answer 23

I don't understand

Answer 24

can you find the slope of AB ?

Answer 25

Yes

Answer 26

what do you get ?

Answer 27

1

Answer 28

close. but that is not the slope for line AB
change in y divided by change in x

Answer 29

can you post what you did?

Answer 30

-1 Sorry wasn't paying attention

Answer 31

yes, that is better.

next, we use this fact
if a line has a slope of m
a line perpendicular to that line will have slope -1/m

(it's one of those facts they expect you to know... to answer this question)

so m= -1
write
-1 divided by m: -1/-1
that simplifies to 1
1 is the slope of axis line.

Answer 32

if you look at your choices, we can cross of the last choice. it has the wrong slope

we have a point (4,3) and a slope m=1
we need to find the equation of a line. Do you know how to do that ?

Answer 33

No...

Answer 34

one way is use
y = m x + b

we know m is 1 so we can write
y = 1x + b
or just
y = x+b

this equation has to work for the point (4,3) (point C)
put in 4 for x and 3 for y:
3= 4 + b
now solve for b
can you do that ?

Answer 35

I would add -4 to both sides, and simplify

Answer 36

b=-1

Answer 37

so m=1 and b=-1 for the axis line:
y = 1x + -1
or
y = x -1 if we pretty it up a little.