In the diagram above, O is the center of the circle. What is the length of chord AC?
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(1) chord BC = 14
(2) the circle has an area of 625π
OA C
Source: Magoosh
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Target question: What is the length of chord AC?BTGmoderatorDC wrote: ↑Thu Sep 16, 2021 11:51 amcpotg_img4300x285.png
In the diagram above, O is the center of the circle. What is the length of chord AC?
(1) chord BC = 14
(2) the circle has an area of 625π
OA C
Source: Magoosh
Given: O is the center of the circle
If O is the center of the circle, then AB is the circle's DIAMETER
If AB is the DIAMETER, then ∠C = 90°, because ∠C is an inscribed angle containing ("holding") the diameter.
So, let's first add this information to the diagram
Statement 1: chord BC = 14
Notice that the length of chord BC has no bearing on the length of chord AC.
In fact, here are two diagrams that satisfy statement 1:
In the lefthand diagram, the answer to the target question is chord AC has length 20
In the righthand diagram, the answer to the target question is chord AC has length 30
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: the circle has an area of 625π
Area of circle = πr²
So, we can write: πr² = 625π
Divide both sides by π to get: r² = 625
Solve: r = 25
So, the circle's radius = 25, which means the DIAMETER AB has length 50.
This time the length of the diameter has little bearing on the length of chord AC.
In fact, here are two diagrams that satisfy statement 2:
In the lefthand diagram, the answer to the target question is chord AC has length 30
In the righthand diagram, the answer to the target question is chord AC has length 40
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined
When we combine the two statements, we see that we know the lengths of two sides of a RIGHT triangle
So, we COULD apply the Pythagorean Theorem to write: 14² + x² = 50²,
And we COULD solve the equation to get x = 48.
However, performing all of those calculations would be a waste of the time, since we need only show that we COULD answer the target question with certainty.
Since we COULD answer the target question with certainty, the combined statements are SUFFICIENT
Answer: C
Cheers,
Brent