Bunuel wrote:

If the circle in the figure above is centered at the origin of the coordinate axes, which of the following coordinates represents a point that lies on the circle?

A. (3,4)

B. (5,5)

C. (1,9)

D. (8,6)

E. (6,6)

lostnumber wrote:

Can someone explain this to me? I understand the radius is 10 and that you could make a right triangle between X and Y, but I don't understand how that helps you find a point on the circle. Wouldn't all points on the arc be outside of the triangle?

lostnumber , if by "arc" you mean the circumference line, no.

A point lies ON a circle if it lies ON the circumference

Points on a circle• any point that lies on the circle will

satisfy the equation for the circle (below)

• any point that lies on the circle will

create a right triangle whose hypotenuse = radius

Basic equation of a circleTo determine whether a point lies on a circle,

plug the x- and y-coordinates into the equation

As

abhimahna and

0akshay0 mention explicitly,

the basic equation of a circle is

\(x^2 + y^2 = r^2\) \(x\) and \(y\) are coordinates

of a point on the circle

\(r\) = radius

Basic equation of circle \(<->\) Pythagorean theoremPythagorean theorem, where c = hypotenuse:

\(a^2 + b^2 = c^2\)Basic equation of circle with center (0,0), r = radius:

\(x^2 + y^2 = r^2\)x replaces a, y replaces b

and r, radius, replaces c, hypotenuse

Any point (x,y) on this circle therefore

satisfies the equation for THIS circle:

\(x^2 + y^2 = 10^2\)

\(x^2 + y^2 = 100\)Scan the answers.

You're looking for x- and y-values

whose squares will sum to 100.

Plug in answers?You can plug in until you find the right pair.

Try A. (3,4)

\(3^2 + 4^2 = r^2\)

\(r^2 = 25\)

\(r = 5\)r must = 10You can keep plugging in

until you get the correct x- and y- values

See the RHS of diagram.*

All points except D lie INSIDE the circle

Their radii lengths are shorter than 10

OR notice: 6-8-10 = 3-4-5 right triangleAnswer D is (8,6)

Radius =

108: 6: 10 = 4x-3x-5x

D's x- and y-coordinates yield lengths

that are a "Pythagorean triplet" triangle

(8, 6) makes a right triangle. See diagram.

Its hypotenuse is the radius.

Check: do D's coordinates (8,6) satisfy equation?Answer D: (8, 6)

\(x^2 + y^2 = r^2\)

\(8^2 + 6^2 = r^2\)

\(64 + 36 = r^2\)

\(r^2 = 100\)

\(r = 10\)That's a match.

Answer D

Hope that helps

* Look at the green triangle on the left.

The green point is (-6, 8). It, too, satisfies the equation

\((-6)^2 + 8^2 = r^2\)

\((36 + 64) = 100 = r^2\)

\(r^2 = 100\)

\(r = 10\)

For more on the equation of a circle, see Bunuel , Circle on a plane , and another site HERE

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