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
Attachment:
T6192ed2.png [ 35.27 KiB | Viewed 230 times ]
_________________
At the still point, there the dance is. -- T.S. Eliot
Formerly genxer123