Can someone please explain this to me step by step since I have a test next Monday and I don’t understand this problem that well
..
1 answer:
You might be interested in
Answer:
14
Step-by-step explanation:
Visualize this situation as in the attachment (it is not scaled).
First, denote by r the radius of the inner circle, R the radius of the outer circle and A the area between both circles. The area of the inner circle is πr² and the area of the outer circle is πR². The area of the inner circle and the area between the circle adds up to the area of the outer circle, that is, πr²+A=πR², then A=πR²-πr².
We are given that A=49π, then 49π=πR²-πr². Divide pi from this equation to obtain 49=R²-r². We will use this later on the problem.
Following the figure, suppose that A is the center of both circles and the chord ED is tangent to the first circle on point C. Construct the triangles ACE and ACD. Both are right triangles because a tangent line is perpendicular to the radius, in this case ED⊥AC.
Now, note that AE=AD=R and AC=r because E,D are points of the outer circle and C is a point of the inner circle. Applying the Pythagorean theorem (on both triangles, we get that CE²=AE²-AC²=R²-r²=49 and CD²=AD²-AC²=R²-r²=49, so that CE=7=CD.
Finally, we compute the length of the chord as ED=EC+CD=CE+CD=7+7=14.
