A circle has a chord of length 10, and the distance from the center of the circle to the chord is 5. What is the area of the circle?
A circle has a chord of length 10, and the distance from the center of the circle to the chord is 5. What is the area of the circle?
Draw the circle and the chord. These don't have to be to scale.
Draw a perpendicular from the center of the circle to the chord.
Draw a diameter from the center of the circle to the end of the chord.
Now you have a right triangle, with one leg is half the chord = 5
The other leg is the distance from the center of the circle to the chord = 5
Use Pythagoras' Theorem to solve for the hypotenuse
and you'll have the diameter of the circle = sqrt(50)
Half the diameter is the radius and the area of the circle is calculated A = π r2
A = 3.1416 • [sqrt(50) / 2]2 = 3.1416 • 50/4 = 3.1416 • 12.5 = 39.27 square units
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CORRECTION...
When you solve for the hypotenuse you'll have the RADIUS not the diameter.
You end up with A = 3.1416 • [sqrt(50)]2 = 3.1416 • 50 = 157.08 units2
Sorry for the clumsy mistake in my original answer above. This is what happens when I try to do it in my head, instead of actually drawing the diagram on paper.
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+14913 A circle has a chord of length 10, and the distance from the center of the circle to the chord is 5. What is the area of the circle?
Hello Guest!
\(r^2=(\frac{chord}{2})^2+(dist.)^2\\ r=\sqrt{(\frac{5}{2})^2+5^2}\)
\(r=\sqrt{\frac{25+100}{4}}=\sqrt{\frac{125}{4}}\)
\(A=\pi r^2=\pi \cdot \frac{125}{4}\)
\(A=98.175\)
\(The\ area\ of\ the\ circle\ is\ 98.175\ .\)
!
