Question

Two circles intersect and have a common chord 16 cm long. The center of the circles are 21 cm apart. The radius of one circle is 10 cm. Find the other radius.

Answer 1

The diagram representing the scenario is shown below

Line AB represents the distance between the centers of the circle. Thus, AB = 21

Line AC represents the radius of one of the circles. Thus, AC = 10

line CD represents the chord whose length is 16. thus, CD = 16

line BC = x represents the radius of the other circle

We can say that line AB divides the chord, CD into two equal halves. This is so because AC = AD = radius

Thus, triangle ACE is a right angle triangle since angle E is 90 degrees

thus, we have

hypotenuse = AC = 10

opposite side = CE = CD/2 = 16/2 = 8

adjacent side = AE

To find AE, we would apply the pythagorean theorem which is expressed as

hypotenuse^2 = opposite side^2 + adjacent side^2

10^2 = 8^2 + AE^2

100 = 64 + AE^2

AE^2 = 100 - 64 = 36

`\begin{gathered} AE\text{ = }\sqrt[]{36} \\ AE\text{ = 6} \end{gathered}`

Recall, AE + EB = 21

6 + EB = 21

EB = 21 - 6

EB = 15

We would apply the pythagorean theorem on triangle BEC. Looking at the triangle,

hypotenuse = CB = x

opposite side = CE = 8

adjacent side = EB = 15

thus, we have

x^2 = 8^2 + 15^2

x^2 = 64 + 225 = 289

`\begin{gathered} x\text{ = }\sqrt[]{289} \\ x\text{ = 17} \end{gathered}`

The other radius is 17 cm