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Arcs and chords

We can use principals from geometry to find the lengths of segments inside circles.  Before we begin, we will state a few theorems.

PYTHAGOREAN THEOREM:  If a and b are two legs of a right triangle, and c is the hypotenuse, then

\({a^2} + {b^2} = {c^2}\)

THEOREM:  If a diameter is perpendicular to a chord, then it bisects the chord and its arcs.

Now we can use these theorems to solve a few problems.

EXAMPLE:  Find the length of the segment x.

Circles-Arcs-and-Chords-1

 

SOLUTION:  We see that x is the leg of a right triangle formed by portions of the diameter, radius, and a chord in the circle.  Since the other leg (9.6) and the hypotenuse (15.6) are known, we can use the Pythagorean Theorem to find x.  We have

\({9.6^2} + {x^2} = {15.5^2}\)
\(92.16 + {x^2} = 240.25\)
\({x^2} = 148.09\)
\(x = 12.2\), rounded to the nearest tenth  

EXAMPLE:  Find the length of the segment x.

Circles-Arcs-and-Chords-2

SOLUTION:  In the picture above, we see that a radius of the circle (and thus the diameter) is parallel to the chord containing the value 15 as one of its segments.  Then, by our second theorem above, the diameter bisects the sword, so that the values of the segments on each side of the diameter are equal.  That is, the value of both segments of the chord are 15.  Thus we have obtained a right triangle with two known legs(6 and 15) and the hypotenuse, x, is the desired value.  We use the Pythagorean Theorem:

\({6^2} + {15^2} = {x^2}\)
\(36 + 225 = {x^2}\)
\(261 = {x^2}\)
\(x = 16.2\),      rounded to the nearest tenth

 

Below you can download some free math worksheets and practice.


Downloads:
4130 x

Find the length of the segment indicated. Round your answer to the nearest tenth if necessary.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Circles-Arcs-and-chords-Easy

Watch bellow how to solve this example:

 

Downloads:
2504 x

Find the length of the segment indicated. Round your answer to the nearest tenth if necessary.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Circles-Arcs-and-chords-Medium

Watch bellow how to solve this example:

 

Downloads:
2969 x

Find the length of the segment indicated. Round your answer to the nearest tenth if necessary.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Circles-Arcs-and-chords-Hard

Watch bellow how to solve this example:

 
 
 

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