If a chord 10 inches long is 5 inches from the center of a circle, find the radius of the circle.

Question

 If a chord 10 inches long is 5 inches from the center of a circle, find the radius of the circle.

Answer 1

A picture would help here, so try to sketch along as I describe the geometrical figure used to solve the problem.

Start with a circle and a chord.  Label the endpoints of the chord A and B.  Now, starting from the center of the circle, draw the bisector of the chord (which itself is a radius of the circle).  Label the center of the circle C, and the intersection of the bisector and the chord D.  Now, we have 4 labelled points.  That's all we need.

 

The chord is 5 inches from the center of the circle; this means that segment CD = 5 inches.  Since AD bisects the chord, which is 10 inches long, we also have segments DA = 5 inches and DB = 5 inches.  The bisector of a chord forms right angles with the chord, so triangles CAD and CBD are both right triangles.  

 

We only need one of these triangles to solve the problem, so let's choose triangle CAD.  CD and AD are the legs of the triangle, and CA is the hypotenuse.  Since CA starts at the center of the circle and ends on the circle itself, it is a radius.  To solve the problem, we need to find the length of CA.  We can represent this length as r (standing for radius) in the solution.

 

Since we have a right triangle, we can use the Pythagorean Theorem:

 

(Leg1)² + (Leg2)² = (hypotenuse)²

(CD)² + (AD)² = (CA)²

5² + 5² = r²

r² = 50

r = √(50)

=√(2*5*5)

=5√2

 

 

 

Answer 2

Draw circle O. Draw chord AB. Draw perpendicular from O to AB, label C.

 

AC = 5, OC = 5, ΔAOC is a right triangle.

 

By Pythagoras, OA = 5√2.

 

The radius is 5√2 inches.

Answer 3

This chord will pass from one side of the circle to the other.  Since it is 5 inches from the center we know it does not pass through the center, and thus does represent the diameter.

 

Draw a circle. Draw a chord through the circle that does not go through the center.  Now draw a straight line from the center of the circle to the center of the chord.  This line is 5 inches long and cuts the chord in half.  now draw a radial line from the center of the circle to the edge of the circle at the same point where an end of the chord touches the edge of the circle.  We now have a right triangle with a 5 inch leg (center to chord midpoint), another 5 inch leg being from the midpoint of the 10 inch long chord to the outer perimeter of the circle.  The radial line is the hypotenuse of this right triangle.

 

Use the pythagorean theorem a²+b²=c²

where a and b are the legs and c is the hypotenuse.

 

5² + 5² = c²

c²=25+25=50

c = √50

c=√(25(2))

c = 5√2

 

This is the radius of the circle.