Up

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:
11002 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:
9369 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:
10695 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:

 
 
 

Facebook PageYouTube Channel

Algebra and Pre-Algebra

Beginning Algebra
Adding and subtracting integer numbers
Dividing integer numbers
Multiplying integer numbers
Sets of numbers
Order of operations
The Distributive Property
Verbal expressions
Beginning Trigonometry
Finding angles
Finding missing sides of triangles
Finding sine, cosine, tangent
Equations
Absolute value equations
Distance, rate, time word problems
Mixture word problems
Work word problems
One step equations
Multi step equations
Exponents
Graphing exponential functions
Operations and scientific notation
Properties of exponents
Writing scientific notation
Factoring
By grouping
Common factor only
Special cases
Linear Equations and Inequalities
Plotting points
Slope
Graphing absolute value equations
Percents
Percent of change
Markup, discount, and tax
Polynomials
Adding and subtracting
Dividing
Multiplying
Naming
Quadratic Functions
Completing the square by finding the constant
Graphing
Solving equations by completing the square
Solving equations by factoring
Solving equations by taking square roots
Solving equations with The Quadratic Formula
Understanding the discriminant
Inequalities
Absolute value inequalities
Graphing Single Variable Inequalities
Radical Expressions
Adding and subtracting
Dividing
Equations
Multiplying
Simplifying single radicals
The Distance Formula
The Midpoint Formula
Rational Expressions
Adding and subtracting
Equations
Multiplying and dividing
Simplifying and excluded values
Systems of Equations and Inequalities
Graphing systems of inequalities
Solving by elimination
Solving by graphing
Solving by substitution
Word problems