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Math Expert V
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In the diagram above, O is the center of the circle. What is the leng  [#permalink]

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Difficulty:   5% (low)

Question Stats: 82% (00:54) correct 18% (01:12) wrong based on 119 sessions

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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 $$625\pi$$

Kudos for a correct solution.

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Re: In the diagram above, O is the center of the circle. What is the leng  [#permalink]

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1

from the question we know that AB is the diameter and the tringle ACB is right tringle 90-60-30

statment 1 insuff where we can not find AC with no information about the other side AB
statment 2 insuff it allow us just to find AB=50 where no information about other side CB
from 1&2 we have AB&CB so we can find AC
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Re: In the diagram above, O is the center of the circle. What is the leng  [#permalink]

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triangle ABC is right angled triangle, as angle made by diameter on a triangle is right angle.
We need any two sides to get the third side`s length.
statement 1 is not sufficient, so is the statement 2.
considering 1 + 2 , We can get the diameter of the circle from 2, which is one side of the triangle and from 1 we have another side.
So, by Pythagoras theorem, we can get length of AC. Sufficient.
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Re: In the diagram above, O is the center of the circle. What is the leng  [#permalink]

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Hi

Statement i >> insufficient

Statement ii >> insufficient
We don't know the length of the other two chords, we just can calculate the diameter

Statement i + ii >> sufficient

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Re: In the diagram above, O is the center of the circle. What is the leng  [#permalink]

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Bunuel wrote:
Attachment:
cpotg_img4-300x285.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 $$625\pi$$

Kudos for a correct solution.

The triangle formed is a right triangle, we just need to figure out 2 of the sides.

Statement 1: Only 1 side is given
Insufficient

Statement 2: Only 1 side is given
Insufficient

Combined, we have 2 sides and can use pythagorean theorem.
Sufficient

Math Expert V
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Posts: 56307
Re: In the diagram above, O is the center of the circle. What is the leng  [#permalink]

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Bunuel wrote: 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 $$625\pi$$

Kudos for a correct solution.

MAGOOSH OFFICIAL SOLUTION:

The fact that AB is a diameter guarantees that angle C = 90º. If we had two sides of right triangle ABC, we could find the third using the Pythagorean Theorem.

Statement #1: this gives us only one side of a right triangle: not helpful. This statement, alone and by itself, is not sufficient.

Statement #2: this allows us to solve for the radius and, hence, the diameter, so we can determine side AB. Nevertheless, this gives us only one side of a right triangle: also not helpful. This statement, alone and by itself, is not sufficient.

Combined statements: We get the length of BC from the first statement, and the length of AB from the second. Now, we have two sides of the right triangle, so we can use the Pythagorean Theorem to solve for the third side, AC. Combined, the statements are sufficient.

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Re: In the diagram above, O is the center of the circle. What is the leng  [#permalink]

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23a2012 wrote:

from the question we know that AB is the diameter and the tringle ACB is right tringle 90-60-30

statment 1 insuff where we can not find AC with no information about the other side AB
statment 2 insuff it allow us just to find AB=50 where no information about other side CB
from 1&2 we have AB&CB so we can find AC

Don't think you can assume it's a 30-60-90 triangle. Flawed reasoning here.
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Re: In the diagram above, O is the center of the circle. What is the leng  [#permalink]

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Bunuel wrote:
Attachment:
cpotg_img4-300x285.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 $$625\pi$$

Kudos for a correct solution.

Target question: What is the length of chord AC?

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 left-hand diagram, the answer to the target question is chord AC has length 20
In the right-hand 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 left-hand diagram, the answer to the target question is chord AC has length 30
In the right-hand 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

Cheers,
Brent

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_________________ Re: In the diagram above, O is the center of the circle. What is the leng   [#permalink] 07 Feb 2019, 12:19
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