DSQ: Calculating the perimeter of an acute / obtuse triangle - GMAT Quantitative Reasoning

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Question

True or false: The perimeter of $\bigtriangleup ABC$ is greater than 50.

Statement 1: $\bigtriangleup ABC$ is an isosceles triangle.

Statement 2: and

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:

$p =\left (AB+ AC \right ) + BC > 28+28 = 56 > 50$ :

The perimeter of $\bigtriangleup ABC$ is greater than 50.

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Question

Find the perimeter of the obtuse $\Delta PGN$ .

I) .

II) ${}PN=\frac{5}6{}GN$ .

Answer

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.

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Question

Give the perimeter of $\bigtriangleup ABC$ .

Statement 1:

Statement 2:

Answer

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

Case 1:

The perimeter of $\bigtriangleup ABC$ is

.

Case 2: .

The perimeter of $\bigtriangleup ABC$ is

.

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

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Question

True or false: The perimeter of $\bigtriangleup ABC$ is greater than 24.

Statement 1:

Statement 2:

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 $\bigtriangleup ABC$ is

, and, since , then

The perimeter of $\bigtriangleup ABC$ is greater than 24.

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Question

True or false: The perimeter of $\bigtriangleup ABC$ is greater than 50.

Statement 1: $\bigtriangleup ABC$ is an isosceles triangle.

Statement 2: and

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.

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Question

Give the perimeter of $\bigtriangleup ABC$ .

Statement 1:

Statement 2:

Answer

The perimeter of $\bigtriangleup ABC$ is equal to the sum of the lengths of the sides; that is, .

From Statement 1 alone, we get

we can add $2 \cdot BC$ to both sides to get

$AB + AC - BC + 2 \cdot BC = 21 + 2 \cdot BC$

$AB + AC + BC = 21 + 2 \cdot BC$

$p= 21 + 2 \cdot BC$

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:

$\underline{AB + AC +2 BC = 45}$

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 :

.

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Question

True or false: The perimeter of $\bigtriangleup ABC$ is greater than 60.

Statement 1: $\bigtriangleup ABC$ is an isosceles triangle.

Statement 2: and

Answer

Assume both statements to be true. An isosceles triangle has two sides of equal length, so either or . By the Triangle Inequality Theorem, the length of each side must be less than the sum of the lengths of the other two; both scenarios are possible, since

and

.

If , the perimeter of $\bigtriangleup ABC$ is

.

If , the perimeter of $\bigtriangleup ABC$ is

.

Without further information, it is impossible to determine whether the perimeter is less than or greater than 60.

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Question

What is the perimeter of $\bigtriangleup ABC$ ?

Statement 1: The triangle with its vertices at the midpoints of $\overline{AB}$ , $\overline{AC}$ , and $\overline{BC}$ has perimeter 34.

Statement 2: , , and are the midpoints of the sides of a triangle with perimeter 136.

Answer

Assume Statement 1 alone. A segment that has as its endpoints the midpoints of two sides of a triangle is a midsegment of the triangle, and its length is half that of the side to which it is parallel. Therefore, the sum of the lengths of the midsegments—that is, the perimeter of the triangle they form, which from Statement 1 is 34, is half the perimeter of the larger triangle, which here is $\bigtriangleup ABC$ . The perimeter of $\bigtriangleup ABC$ is therefore 68.

Assume Statement 2 alone. Here, $\bigtriangleup ABC$ itself is the triangle formed by the midsegments. Since the larger triangle has perimeter 136, $\bigtriangleup ABC$ has perimeter half this, or 68.

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Question

True or false: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ have the same perimeter.

Statement 1: $\bigtriangleup ABC$ is isosceles and $\bigtriangleup DEF$ is scalene.

Statement 2: and .

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. $\bigtriangleup ABC$ is isosceles, so two of its sides have equal length; however, it cannot hold that $\overline{AB} \cong \overline{BC}$ ; if so, then, since $\overline{AB} \cong \overline{DE}$ and $\overline{BC}\cong \overline{EF}$ , it would follow that $\overline{DE}\cong \overline{EF}$ , which contradicts $\bigtriangleup DEF$ being scalene. Therefore, either $\overline{AC} \cong \overline{AB}$ or $\overline{AC} \cong \overline{BC}$ . If $\overline{DF} \cong \overline{AC}$ , then $\overline{DF}$ is congruent to one other side of $\bigtriangleup ABC$ , and, consequently, one other side of $\bigtriangleup DEF$ , contradicting $\bigtriangleup DEF$ being scalene. Therefore, ; as stated before, the perimeters are equal if and only if , so the perimeters are not equal.

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Question

Given $\bigtriangleup ABC$ and Square , which one has the greater perimeter?

Statement 1:

Statement 2: $BC = 2 \cdot QR$

Answer

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 $\overline{SU}$ and $\overline{QR}$ is $\sqrt{2}$ times the length of a side, or simply $\sqrt{2}$ .

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: $BC = 2 \sqrt{2}$

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, $\bigtriangleup ABC$ has greater perimeter than Square .

Assume Statement 2. By similar reasoning, since one side has length $2 \sqrt{2}$ , the perimeter is at greater than twice this, or $4 \sqrt{2}$ , which is greater than 4, so $\bigtriangleup ABC$ has greater perimeter than Square .

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Question

Given Triangle $\bigtriangleup ABC$ and Square , which one has the greater perimeter?

Statement 1:

Statement 2:

Answer

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 $\overline{SU}$ and $\overline{QR}$ is $\sqrt{2}$ times the length of a side, or simply $\sqrt{2}$ .

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: $AB = \sqrt{2}$

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:

$AC < 1+ \sqrt{2}$

Also,

$\sqrt{2} -1 < AC$

So,

$-1 + \sqrt{2} < AC < 1+ \sqrt{2}$

Add and to all three expressions; the expression in the middle is the perimeter of $\bigtriangleup ABC$ :

$-1 + \sqrt{2} + AB + BC < AC + AB + BC< 1+ \sqrt{2} + AB + BC$

$-1 + \sqrt{2} + \sqrt{2 }+ 1< p< 1+ \sqrt{2} + \sqrt{2 }+ 1$

$2 \sqrt{2} < p< 2+ 2\sqrt{2}$

Since $\sqrt{2} \approx 1.4$ , 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.

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Question

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , with $\overline{AB} \cong \overline{DE}$ and $\angle A \cong \angle D$ .

True or false: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ have the same perimeter.

Statement 1: $\overline{BC} \cong \overline{EF}$

Statement 2: $\angle B \cong \angle E$

Answer

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

Case 1: $\bigtriangleup ABC \cong \bigtriangleup DEF$ . By definition, $\overline{AB} \cong \overline{DE}$ and $\angle A \cong \angle D$ , satisfying the condtions of the main body of the problem, and $\overline{BC} \cong \overline{EF}$ , 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 $\overline{AB} \cong \overline{DE}$ , $\angle A \cong \angle D$ (their being the same segment and angle, respectively, in the diagram), and $\overline{BC} \cong \overline{EF}$ by construction. Note, however, that , so:

.

Making the perimeters different.

Now assume Statement 2 alone. $\overline{AB}$ is the included side of $\angle A$ and $\angle B$ , and $\overline{DE}$ is the included side of $\angle D$ and $\angle E$ . The three congruence statements given in the main body and Statement 2 together set the conditions of the Angle-Side-Angle Postulate, so $\bigtriangleup ABC \cong \bigtriangleup DEF$ , and the perimeters are indeed the same.

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Question

What is the perimeter of $\bigtriangleup ABC$ ?

Statement 1: and

Statement 2: $\angle B \cong \angle C$

Answer

From Statement 1 alone, we know that two sides are of length 10, but we are not given any clue as to the length of the third. Therefore, we cannot calculate the sum of the side lengths, which is the perimeter. Statement 2 alone is unhelpful, since it only gives that two angles have equal measure; applying the Converse of the Isosceles Triangle Theorem, it can be determined that their opposite sides $\overline{AC}$ and $\overline{AB}$ are congruent, but no actual side lengths can be found.

Now assume both statements to be true. From Statement 1, $\overline{AB} \cong \overline{BC}$ , and, as stated before, it follows from Statement 2 that $\overline{AB} \cong \overline{AC}$ . Therefore, the triangle is equilateral with sides of common length 10, making the perimeter 30.

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