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GMAT/Data Sufficiency - Chord Query
Chord Query
For a circle with center point P, chord XY is the perpendicular bisector of radius AP (A is a point on the edge of the circle). What is the length of cord XY?
(1) The circumference of circle P is twice the area of circle P.
(2) The length of Arc XAY =  .
(A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) Each statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient.
| ОТВЕТЫ И КОММЕНТАРИИ |
| Решение |
Since no picture is given in the problem, draw it. Below, find the given circle with center P and chord XY bisecting radius AP.
Although, the picture above is helpful, drawing in an additional radius is often an important step towards seeing the solution. Thus, we will add to the picture by drawing in radius XP as shown below.
Answer
Since no picture is given in the problem, draw it. Below, find the given circle with center P and chord XY bisecting radius AP.
Although, the picture above is helpful, drawing in an additional radius is often an important step towards seeing the solution. Thus, we will add to the picture by drawing in radius XP as shown below.
Since XY bisects radius AP at point B, segment BP is half the length of any radius.
Since BP is half the length of radius XP, right triangle XBP must be a 30-60-90 triangle with sides in the ratio of  .
Therefore, finding the length of any side of the triangle, will give us the lengths of the other two sides.
Finding the length of radius XP will give us the length of XB, which is half the length of chord XY. Thus, in order to answer the question--what is the length of chord XY?--we need to know only one piece of information:
The length of the radius of the circle.
Statement (1) alone tells us that the circumference of the circle is twice the area of the circle. Using this information, we can set up an equation and solve for the radius as follows:
Therefore Statement (1) alone is sufficient to answer the question.
Statement (2) alone tells us that the length of Arc XAY =  .
Arc XAY is made up of arc XA + arc AY.
Given that triangle XBP is a 30:60: 90 triangle, we know that <XPA = 60 degrees and can deduce that <APY = 60 degrees as well. Therefore Arc XAY =120 degrees or 1/3 of the circumference of the circle. Using this information, we can solve for the radius of the circle by setting up an equation as follows:
Therefore, Statement (2) alone is also sufficient to answer the question. The correct answer is D, each statement ALONE is sufficient.
Since XY bisects radius AP at point B, segment BP is half the length of any radius.
Since BP is half the length of radius XP, right triangle XBP must be a 30-60-90 triangle with sides in the ratio of  .
Therefore, finding the length of any side of the triangle, will give us the lengths of the other two sides.
Finding the length of radius XP will give us the length of XB, which is half the length of chord XY. Thus, in order to answer the question--what is the length of chord XY?--we need to know only one piece of information:
The length of the radius of the circle.
Statement (1) alone tells us that the circumference of the circle is twice the area of the circle. Using this information, we can set up an equation and solve for the radius as follows:
Therefore Statement (1) alone is sufficient to answer the question.
Statement (2) alone tells us that the length of Arc XAY =  .
Arc XAY is made up of arc XA + arc AY.
Given that triangle XBP is a 30:60: 90 triangle, we know that <XPA = 60 degrees and can deduce that <APY = 60 degrees as well. Therefore Arc XAY =120 degrees or 1/3 of the circumference of the circle. Using this information, we can solve for the radius of the circle by setting up an equation as follows:
Therefore, Statement (2) alone is also sufficient to answer the question. The correct answer is D, each statement ALONE is sufficient.
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Delegates to the Republican Party convention chose their candidate and was able
to ratify all aspects of his campaign platform at the convention in Denver.
А) chose their candidate and was able
В) chose their candidate and were able
С) chose its candidate and were able
D) chose its candidate and was able
Е) had chosen their candidate and was able
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Вверх 
