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?

Guest Jun 23, 2020

#1**0 **

*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 r^{2}

A = 3**.**1416 • [sqrt(50) / 2]^{2} = 3**.**1416 • 50/4 = 3**.**1416 • 12**.**5 = **39.27 square units**

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Guest Jun 23, 2020

#2**0 **

**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 units ^{2}**

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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Guest Jun 23, 2020

#3**0 **

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!**

\(r^2=(\frac{chord}{2})^2+(dist.)^2\\ r=\sqrt{(\frac{5}{2})^2+5^2}\)

\(r=\sqrt{\frac{25+100}{4}}=\sqrt{\frac{125}{4}}\)

\(A=\pi r^2=\pi \cdot \frac{125}{4}\)

\(A=98.175\)

\(The\ area\ of\ the\ circle\ is\ 98.175\ .\)

!

asinus Jun 23, 2020