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# help

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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?

Jun 23, 2020

#1
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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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Jun 23, 2020
#2
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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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Guest Jun 23, 2020
#3
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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!

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

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

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

$$A=98.175$$

$$The\ area\ of\ the\ circle\ is\ 98.175\ .$$ !

Jun 23, 2020
edited by asinus  Jun 23, 2020
edited by asinus  Jun 23, 2020
#6
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asinus ... the length of the chord is 10 ... so the expression under the radical in the second step should

be (ten over two) squared ... instead of (five over two) squared.

Guest Jun 23, 2020
#4
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Area is   50*pi = 157.0796327

Jun 23, 2020
#5
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r = sqrt( 2 * 52 ) = 7.071067718

A  = r2 pi =  50*pi = 157.0796327 units squared Jun 23, 2020