# TANCET 2014 DS 67: Geometry

## Directions

The question is followed by two statements labeled (1) and (2) in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the problem plus your knowledge of mathematics and everyday facts, choose the answer as:

1. Choice 1 if statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Choice 2 if statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. Choice 3 if both the statements (1) and (2) TOGETHER are sufficient, but NEITHER statement alone is sufficient.
4. Choice 4 if each statement ALONE is sufficient.
5. Choice 5 if statements (1) and (2) TOGETHER are not sufficient, and additional data is needed.

## Question

Easy Data Sufficiency

If point X is directly north of point Y and directly west of point Z, what is the distance from point X to point Z?

1. Statement 1: The distance from Y to Z is 20 cm.
2. Statement 2: The distance from X to Y is equal to half the distance from Y to Z.

Correct Answer    Choice (3). Statements (1) and (2) TOGETHER are sufficient.

## Explanatory Answer - step by step

• ### What should we know from the Question Stem?

Before evaluating the two statements, answer the following questions to get clarity on when the data is sufficient.

#### What kind of an answer will the question fetch?

The question is "What is the distance from point X to point Z?"

The answer to the question will the measure of the distance between point X and point Z. It is essentially a number followed by a unit of distance - which in this question is cm.

#### When is the data sufficient?

If we are able to come up with a unique value for the distance between X and Z, the data is sufficient.

If we are either not able to find an answer or if we are not able to find a unique value, the data is NOT sufficient.

#### What additional information do we have from the question stem? If point X is directly north of point Y and directly west of point Z. The adjacent figure shows how the 3 points lie in the x-y plane.

Because the point X is directly north of Y and directly west of Z, the three points form the vertices of a right triangle that is right angled at point X.

• ### Statement (1) ALONE

#### The distance from Y to Z is 20 cm. The distance from Y to Z provides the length of the hypotenuse of the right triangle. We will not be able to find the distance from X to Z.

Statement (1) ALONE is NOT sufficient.

If statement (1) ALONE is NOT sufficient, we can eliminate choices 1 and 4.

Choices narrow down to 2, 3, or 5.

• ### Statement (2) ALONE

#### The distance from X to Y is equal to half the distance from Y to Z.

Remember: When you are evaluating statement (2) ALONE, please do not recall information that you read in statement (1). Any information from statement (1) should not be used while evaluating statement (2).

We know the ratio of the lengths of the two sides of the right triangle.

We can infer that the distance from X to Z will be $$sqrt { 3 } \\$ times the distance from X to Y. This information is not enough to find the exact distance form X to Z. We could NOT find the answer to the question with the information given in statement$2).

Statement (2) ALONE is NOT sufficient.

Hence, we can eliminate choice (2).

Answer narrrows down to choice 3 or choice 5.

• ### Combine the Statements

+

#### The distance from X to Y is equal to half the distance from Y to Z. We can deduce that the distance from X to Y = 10 cm.

We need to find the distance from X to Z. It happens to be the length of one of the perpendicular sides of the right triangle.

We know the measure of the remaining two sides. The hypotenuse measures 20. Side YZ measures 10.

Therefore, length of XZ = $$sqrt { { 20 }^{ 2 }-{ 10 }^{ 2 } } =10\sqrt { 3 } \\$ Using statements$1) and (2) TOGETHER, we could find a unique answer for the distance between points X and Z.

We could answer the question with a DEFINITE Yes using the information given in statement (2).

Hence, the correct answer is Choice 3. Both statements are needed to answer the question.

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