The circular base of an above-ground swimming pool lies in

The circular base of an above-ground swimming pool lies in a level yard and just touches two straight sides of a fence at points A and B, as shown in the figure above. Point C is on the ground where the two sides of the fence meet. How far from the center of the pool's base is point A?

Consider the diagram below:

Notice that we are asked to find the length of QA, or the radius of the circular base.

(1) The base has area 250 square feet --> \(area=\pi{r^2}=250\) --> we can find r. Sufficient.

(2) The center of the base is 20 feet from point C. Triangle CQA IS a right triangle, because the tangent line (CA) is always at the 90 degree angle (perpendicular) to the radius (QA) of a circle.

So, we have that CQ=hypotenuse=20. BUT, knowing that hypotenuse equals to 20 DOES NOT mean that the sides of the right triangle necessarily must be in the ratio of Pythagorean triple - 12:16:20. Or in other words: if \(x^2+y^2=20^2\) DOES NOT mean that \(x=12\) and \(y=16\). Certainly this is one of the possibilities but definitely not the only one. In fact \(x^2+y^2=20^2\) has infinitely many solutions for \(x\) and \(y\) and only one of them is \(x=12\) and \(y=16\).

For example: \(x=1\) and \(y=\sqrt{399}\) or \(x=2\) and \(y=\sqrt{396}\)...

So, this statement is not sufficient to get QA.

Answer: A.


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IMO A,

Q How far from the center of the pool's base is point A? i.e. we have to find the radius of the pool

1) area = 250 sq feet. we can use pi(r^2)=250 to find the radius of the pool hence A is sufficient

2) let the center be o, we have OC=20, by using Pythagoras theorem in right angle triangle OCA, (as CA is tangent to the circular base )

we have (20)^2= (OA)^2+(CA)^2
this equation can have infinitely many solutions hence 2 alone is not sufficient

Basically, we need to find the radius of the given circle.

St1: The base has area 250 square feet. Sufficient.

St2: The center of the base is 20 feet from point C.

Let the center be O. This implies OC = 20. Since a tangent is perpendicular with the radius of a circle, we will have triangle OAC as a right triangle, right angled at A. OC is the hypotenuse. Even using Pythagoras theorem, we cannot get a definite value for the radius. Not sufficient.

Answer (A).

We can see that the distance of the center of the pool's base, O and the point A is nothing but the radius of the circular base. Hence, we have to find find the radius of the base.

Statement 1: We know the area of the circular base.
Hence, we can determine the radius of the base.

Sufficient

Statement 2: OC = 20
This doesn't tell us anything about the radius.

Not sufficient

The correct answer is A.

Hi Bunuel,

For the sake of argument(I know that this doesn't provide any additional value to help solve this problem) -- how can we assume that BCQ and ACQ are equal? I realize that both lines are tangent to the circle but doesn't that mean that the line is perpendicular to the center of the circle. Theoretically, that has infinite points, doesn't it?

Thanks

Two important properties:
1. The tangent line is always at the 90 degree angle (perpendicular) to the radius of a circle.
2. Tangent segments to a circle from the same external point are congruent.

Hope it helps.
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Hi Bunuel,

A few follow up questions to statement 2:

1) If we know that the segments BC and AC are equal (one of the properties of a tangent line that intersects a circle), can't we assume that the angle QAC is a 45.45.90 ? I would imagine that since QA = QB and since they are both 90, they will intersect the circle symmetrically and therefore create a 45 45 90. Am I inferring too much here?

2) If we know that hypotenuse is 20, I was under the impression that a 20 hyp and a 90 degree angle ALWAYS results in a 12:16:20?

Thanks!

1. Yes, CA = CB. You are right about it. But this does not mean that CQA or CQB are 45-45-90 triangles. Consider this, by moving point C triangles CQA and CQB change, so only one position of C would give 45-45-90 triangles, while other positions of C will give other types of right triangles.

2. This is explained in the solution. Knowing that hypotenuse is 20 does not necessarily mean that we have a 12:16:20 right triangle. For more on this trap check the following questions:
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what-is-the-area-of-rectangular-region-r-166186.html
if-distinct-points-a-b-c-and-d-form-a-right-triangle-abc-129328.html

Hope this helps.
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If the question had a different context, and it said the sides had to be integers, would statement 2 be sufficient ?

Yes by declaring the sides are integers only we will eliminate all other possible combinations and will be left with only integer combination of 12 and 16,which gives 20 as hypotenuse.

The goal of the problem is to find the distance between the center of the pool to point A, a point in the edge of the pool. In other words, what is the radius of the pool?

Statement 1) The base has area 250 square feet.

A = pi*R^2

250 = pi*R^2

Sufficient.

Statement 2) Center of the base is 20 feet from point C.

Whenever a point is tangent to a circle, it is perpendicular to the radius. In this case, it creates a right triangle with hypotenuse 20 from C to O, the center of the circle. However, we don't know the dimensions of the at least one of the other sides, so we cannot find the radius. Insufficient.

Wanted: radius?

1) Area given => Can get radius
Sufficient

2) Let centre be at O, then OC = 20

r^2 + CB^2 = 400
Don't know CB
Not sufficient

ANSWER: A

I have a question about statement 2. If we build polygon OABC, then sides OA and OB are equal since both of them are radiuses and angles CBO and CAO are 90 degrees , therefore OABC is a square. Then CO is diagonal and equals square root of ( 2*(one side of square^2) = 20 and statement 2 is sufficient. Or does the property two adjacent sides equal - polygon is square doesn't apply to all polygons?

2020prep2020
There is a problem with the highlighted part.
Surely it is one of the possibilities but not the only one.

Reason: Angle C and O sum to 180 and they have many possible pair of values, which is what we don't want to reach a unique solution. Because of this CA/OA and CB/OB vary, leading to again many possibilities.

Hope this helps.
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The circular base of an above-ground swimming pool lies in a level yard and just touches two straight sides of a fence at points A and B, as shown in the figure above. Point C is on the ground where the two sides of the fence meet. How far from the center of the pool's base is point A?

(1) The base has area 250 square feet.

This statement gives us the area of the circular base, from which we can determine the radius. SUFFICIENT.

(2) The center of the base is 20 feet from point C.

If we let the center of the circle be O, then OC equals 20. However, we're not able to determine the value of OA. There are MANY possibilities -- simply knowing the hypotenuse is not enough for us to determine the sides of the triangle. INSUFFICIENT.

Answer is A.
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Let me know if I am correct. If it was stated that the numbers are integers(negative cannot be possible here) then we could have opted for option D, right?

Solution:

We need to determine the distance between the center of the pool’s base and point A. If we let O be the center of the base, we need to determine the length of OA. Notice that OA is the radius of the pool.

Statement One Alone:

Since we know the area of the pool’s base, we can determine its radius. Statement one alone is sufficient.

Statement Two Alone:

We are given that OC = 20 ft, which is the length of the hypotenuse of the right triangle OAC. However, since we don’t know the length of AC, we can’t determine OA. Statement two alone is not sufficient.

Answer: A
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Statement (1) is not sufficient to answer the question.

The area of a circle is given by πr², where r is the radius of the circle. So we know that 250 = πr². But this equation only gives us one possible value of r, namely r = 10. There are infinitely many other circles with an area of 250 square feet, each with a different radius. Therefore, statement (1) alone is not sufficient to determine the distance between point A and the center of the pool's base.

Statement (2) is sufficient to answer the question.

Triangle CQA is a right triangle, because the tangent line (CA) is always at the 90 degree angle (perpendicular) to the radius (QA) of a circle. Since we are given that the center of the base is 20 feet from point C, we know that the hypotenuse of triangle CQA is 20 feet. We also know that the area of the circle is 250 square feet, which means that the radius of the circle is 10 feet. Therefore, the leg opposite angle A (which is the distance between point A and the center of the pool's base) is √(20² - 10²) = √225 = 15 feet.

Therefore, the answer is (2) for above ground pool installation .