IXL - Derive equations of circles using the Pythagorean theorem (Geometry practice)

Use the Pythagorean theorem to derive the equation of the circle centered at

(

–

)

with a radius of 3.

To which triangle should you apply the Pythagorean theorem?

Complete the paragraph.

The length of the triangle's horizontal leg is . The length of its vertical leg is . The length of its hypotenuse is . By the Pythagorean theorem, the equation of the circle is .

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Learn with an example

question

Use the Pythagorean theorem to derive the equation of the circle centered at

(

–

)

with a diameter of 10.

To which triangle should you apply the Pythagorean theorem?

Complete the paragraph.

The length of the triangle's horizontal leg is . The length of its vertical leg is . The length of its hypotenuse is . By the Pythagorean theorem, the equation of the circle is .

solution

The center of the circle is

(

–

)

Since the diameter is 10, the radius is 5. So, the equation of the circle defines the set of points

(

)

that are all 5 units away from

(

–

)

This diagram shows one possible location for

(

)

You can use the Pythagorean theorem to derive the circle's equation. First, draw a right triangle inside the circle. Use the circle's radius as the triangle's hypotenuse.

Find the lengths of the triangle's sides. Since

(

)

can be any point on the circle, use absolute values to find the lengths.

The
x
-coordinates
of the endpoints of the horizontal leg are
x
and 1. So, its length is
absolute value bar
x
–
1
absolute value bar
.
The
y
-coordinates
of the endpoints of the vertical leg are
y
and
–
6
.
So, its length is
absolute value bar
y
–
(
–
6
)
absolute value bar
=
absolute value bar
y
+
6
absolute value bar
.
The hypotenuse is a radius. So, its length is 5.

Substitute the side lengths into the Pythagorean theorem to write the equation of the circle.

(

–

)

(

)

Simplify the equation. Use the fact that squaring the absolute value of a number is the same as squaring the number. For example, both

(

–

)

and

(

–

)

equal 9.

(

–

)

(

)

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Work it out

Not feeling ready yet? These can help:

P.1Pythagorean theorem

Y.2Find the radius or diameter of a circle

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