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## Example Questions

### Example Question #1 : Dsq: Calculating The Perimeter Of An Acute / Obtuse Triangle

Find the perimeter of the obtuse .

I) .

II) .

**Possible Answers:**

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Both statements are needed to answer the question.

Neither statement is sufficient to answer the question. More information is needed.

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Either statement is sufficient to answer the question.

**Correct answer:**

Neither statement is sufficient to answer the question. More information is needed.

We are told PGN is obtuse, so it has one angle larger than 90 degrees. However, we don't know what that angle is. To find the perimeter we need all three sides.

I) Relates the two shorter sides.

II) Relates the longest side to one of the short sides.

However, we cannot find any of our side lengths, so we cannot find the perimeter.

### Example Question #2 : Dsq: Calculating The Perimeter Of An Acute / Obtuse Triangle

Give the perimeter of .

Statement 1:

Statement 2:

**Possible Answers:**

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

The perimeter of is equal to the sum of the lengths of the sides; that is, .

From Statement 1 alone, we get

we can add to both sides to get

However, without any further information, we cannot determine the actual perimeter.

A similar argument shows that Statement 2 alone gives insufficient information as well.

However, suppose we were to multiply both sides of the equation in Statement 1 by 2, then add both sides of Statement 2:

Divide both sides by 3:

Since

,

we can substitute 29 for and find :

While we cannot find or individually, this is not necessary; in the perimeter formula, we can substitute 29 for and 8 for :

.

### Example Question #3 : Dsq: Calculating The Perimeter Of An Acute / Obtuse Triangle

True or false: and have the same perimeter.

Statement 1: is isosceles and is scalene.

Statement 2: and .

**Possible Answers:**

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

Statement 1 alone provides insufficient information to answer the question, since it is possible for an isosceles triangle, which has two or three sides of equal length, to have perimeter equal to or not equal to a scalene triangle, which has three sides of different lengths. For example, a triangle with sides of length 10, 10, and 12 has perimeter , the same as a triangle with sides of length 9, 10, and 13, since , but a triangle with sides of length 10, 10, and 13 has perimeter .

Statement 2 alone provides insufficient information to answer the question. Since and , it follows that the perimeter are equal if and only if ; we are not told whether this is true or false.

Now assume both statements. is isosceles, so two of its sides have equal length; however, it cannot hold that ; if so, then, since and , it would follow that , which contradicts being scalene. Therefore, either or . If , then is congruent to one other side of , and, consequently, one other side of , contradicting being scalene. Therefore, ; as stated before, the perimeters are equal if and only if , so the perimeters are not equal.

### Example Question #4 : Dsq: Calculating The Perimeter Of An Acute / Obtuse Triangle

True or false: The perimeter of is greater than 24.

Statement 1:

Statement 2:

**Possible Answers:**

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

Statement 1 alone gives insufficient information. By the Triangle Inequality Theorem, the sum of the lengths of the shortest two sides of a triangle must be greater than the length of the longest. Examine these two scenarios:

Case 1:

This triangle satisfies the triangle inequality, since ; its perimeter is

Case 2:

This triangle satisfies the triangle inequality, since ; its perimeter is .

Therefore, Statement 1 alone does not answer whether the perimeter is less than, equal to, or greater than 24.

Assume Statement 2 alone. Again, ; since, by Statement 2, , by substitution, . The perimeter of is

, and, since , then

The perimeter of is greater than 24.

### Example Question #5 : Dsq: Calculating The Perimeter Of An Acute / Obtuse Triangle

Given and Square , which one has the greater perimeter?

Statement 1:

Statement 2:

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

For the sake of simplicity, we will assume the length of each side of the square is 1; this reasoning works independently of the sidelength. The perimeter of the square is, as a result, 4, and the length of each of the diagonals and is times the length of a side, or simply .

The equivalent question becomes whether the perimeter of the triangle is greater than, equal to, or less than 4. The statements can be rewritten as

Statement 1: - or equivalently,

Statement 2:

Assume Statement 1 alone. By the Triangle Inequality, the sum of the lengths of two sides of a triangle must exceed the third. Therefore,

and

Since from Statement 1, , t

By the Addition Property of Inequality, we can add to both sides;

The perimeter of the triangle is greater than 4; equivalently, has greater perimeter than Square .

Assume Statement 2. By similar reasoning, since one side has length , the perimeter is at greater than twice this, or , which is greater than 4, so has greater perimeter than Square .

### Example Question #2 : Dsq: Calculating The Perimeter Of An Acute / Obtuse Triangle

True or false: The perimeter of is greater than 50.

Statement 1: is an isosceles triangle.

Statement 2: and

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

Statement 1 is insufficient, as it only gives that two sides are of equal length; it gives no side lengths, nor does it give any measurements that yield the side lengths.

Assume Statement 2 alone. By the Triangle Inequality, the length of each side must be less than the sum of the lengths of the other two, so

Also,

Therefore,

,

and we can find the range of the values of the perimeter

by adding:

Therefore, the perimeter may or may not be greater than 50.

Assume both statements to be true. An isosceles triangle has two sides of the same length, so either or .

If , the perimeter is

If , the perimeter is

Either way, the perimeter is less than 50.

### Example Question #12 : Acute / Obtuse Triangles

Given Triangle and Square , which one has the greater perimeter?

Statement 1:

Statement 2:

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

For the sake of simplicity, we will assume the length of each side of the square is 1; this reasoning works independently of the side length. The perimeter of the square is, as a result, 4, and the length of each of the diagonals and is times the length of a side, or simply .

The equivalent question becomes whether the perimeter of the triangle is greater than, equal to, or less than 4. The statements can be rewritten as

Statement 1:

Statement 2:

We show that these two statements together provide insufficient information.

By the Triangle Inequality, the sum of the lengths of two sides of a triangle must exceed the third. We can get the range of values of using this fact:

Also,

So,

Add and to all three expressions; the expression in the middle is the perimeter of :

Since , for all practical purposes,

Therefore, we cannot tell whether the perimeter is less than, equal to, or greater than 4. Equivalently, we cannot determine whether the triangle or the square has the greater perimeter.

### Example Question #7 : Dsq: Calculating The Perimeter Of An Acute / Obtuse Triangle

Give the perimeter of .

Statement 1:

Statement 2:

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

We demonstrate that both statements together provide insufficient information by examining two cases:

Case 1:

The perimeter of is

.

Case 2: .

The perimeter of is

.

Both cases satisfy the conditions of both statements but different perimeters are yielded.

### Example Question #3 : Dsq: Calculating The Perimeter Of An Acute / Obtuse Triangle

True or false: The perimeter of is greater than 50.

Statement 1: is an isosceles triangle.

Statement 2: and

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

Statement 1 is insufficient, as it only gives that two sides are of equal length; it gives no side lengths, nor does it give any measurements that yield the side lengths.

Assume Statement 2 alone. By the Triangle Inequality, the length of each side must be less than the sum of the lengths of the other two, so

We can find the minimum of the perimeter:

:

The perimeter of is greater than 50.

### Example Question #4 : Dsq: Calculating The Perimeter Of An Acute / Obtuse Triangle

Given: and , with and .

True or false: and have the same perimeter.

Statement 1:

Statement 2:

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

Assume Statement 1 alone. We show that this provides insufficient information by examining two scenarios.

Case 1: . By definition, and , satisfying the condtions of the main body of the problem, and , satisfying the condition of Statement 1. Since the triangles are congruent, all three pairs of corresponding sides have the same length, so the perimeters are equal.

Case 2: Examine this diagram, which superimposes the triangles such that and coincide with and , respectively:

The conditions of the main body and Statement 1 are met, since , (their being the same segment and angle, respectively, in the diagram), and by construction. Note, however, that , so:

.

Making the perimeters different.

Now assume Statement 2 alone. is the included side of and , and is the included side of and . The three congruence statements given in the main body and Statement 2 together set the conditions of the Angle-Side-Angle Postulate, so , and the perimeters are indeed the same.

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