Two circles intersect and have a common chord 24 cm long. The centers of the circles are 21 cm apart. The radius of one circle is 13 cm. Find the radius of the other circle.
I don't get it. Why won't the radii of the two circles be the same? If there's a way to figure it out, please don't do the math, just tell me how to do it....
Just draw a rough sketch: two intersecting circles.Originally Posted by Itachi888Uchiha
Draw the chord -- the line between the two intersections.
Next drawn a line between the two radius points.
It does not need to be accurate to scale. Freehand two circles & two lines.
Then drawn a line from one of the circle intersection points to the center of one of the circles. Call that distance 13 -- the known radius.
You now have a triangle:
1. center of circle to an intersection point.
2. intersection point to 1/2 the chord, or where the chord is intersected by the line connecting the two radii.
3. from the mid point of the chord back to the center of the circle with known radius.
Call the distance between the two circle centers, which is 21, as x, and 21-x.
You now have three sides of a right triangle.
x^2 + (24/2)^2 = 13^2. You can solve for x.
Then use that info to work on the other (unknown) radius.
Why would they be the same...?
Draw two overlapping circles, with one larger than the other. (The particular scale doesn't really matter. The picture is just a place-holder.) In my drawing, the larger circle is on the left.
Draw the line (the "chord") between the two intersection points. Draw the line, perpendicular to this, between the two centers. Note that this second line must split the chord in half.
Draw one radius line from each center to one of the intersection points. In my drawing, I draw the radii up to the upper of the two points. Label one radius line as "13" and the other as "r". In my drawing, I labelled the left-hand circle's radius as "13".
Note that you now have two right triangles. They share a height line of length 12 (being half of the chord's length), and they have hypotenuse values of 13 and r.
The center-to-center line forms the bases of the two triangles. Label one base as "x" and the other as "y". In my picture, I labelled the left-hand triangle's base as "x".
Apply the Pythagorean Theorem to the triangle with sides x, 12, and 13. Solve for x.
Noting the x + y = 21, use the value for x to find the value for y.
Using this value for y, along with the Pythagorean Theorem, find the value of r.
If you get stuck, please reply showing how far you have gotten. Thank you!
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