DSQ: Calculating the length of the side of an equilateral triangle - GMAT Quantitative Reasoning

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Question

Is $\Delta ABC$ an equilateral triangle?

Statement 1: $m\angle A =m\angle B = 60^{\circ }$

Statement 2: $\Delta ABC \sim \Delta DEF$ , and $\Delta DEF$ is equiangular.

Answer

If $m\angle A =m \angle B = 60^{\circ }$ , then

$m\angle C = 180 - \left (m\angle A + m \angle B \right )=180 - \left (60 +60 \right ) = 60^{\circ }$ .

This makes $\Delta ABC$ an equiangular triangle.

If $\Delta ABC \sim \Delta DEF$ , and $\Delta DEF$ is equiangular, then, since corresponding angles of similar triangles are congruent, $\Delta ABC$ has the same angle measures, and is itself equiangular.

From either statement, since all equiangular triangles are equilateral, we can draw this conclusion about $\Delta ABC$ .

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Question

True or false: $\Delta ABC$ is equilateral.

Statement 1: The perimeter of $\Delta ABC$ is .

Statement 2: .

Answer

The two statements together provide insufficient information. A triangle with sides , , and is equilateral and has perimeter ; A triangle with sides , , and is not equilateral and has perimeter .

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Question

$\overline{DC}$ is the height of $\bigtriangleup ABC$ . What is the length of $\overline{AB}$ ?

(1) $\overline{DC}=4$

(2) $\overline{DB}=3$

Answer

To find the answer we should know more about the characteristics of the triangle, i.e. its angles, sides...

Statement 1 alone is obviously insufficient, since we don't know whether the triangle is equilateral, nothing can be said about AB.

Statement 2 is equally as unhelpful as statement 1, since we don't know whether ABC is of a specific type of triangle.

Taken together, these statements allow us to calculate the length of CB, but we can't go further, because we don't know what is AD.

Therefore statements 1 and 2 are not sufficient even taken together.

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Question

ABC is an equilateral triangle inscribed in the circle. What is the length of side AB?

(1) The area of the circle is $16\pi$

(2) The perimeter of triangle ABC is $\frac{36}{\sqrt{3}}$

Answer

To find the length of the side, we would need to know anything about the lengths in the circle or in the triangle.

From statement 1, we can find the radius of the circle, which allows us to calculate the height of the triangle, since the radius is $\frac{2}{3}$ of the height. And finally since the triangle is equilateral, we can also calculate the length of the sides from the height.

Therefore statement 1 is sufficient.

Statement 2 also gives us useful information, indeed the perimeter is simply three times the length of the sides.

Therefore the final answer is each statement alone is sufficient.

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Question

Find the side length of $\Delta FHT$ .

I) $\Delta FHT$ has perimeter of .

II) $\measuredangle F$ is equal to $\measuredangle T$ which is $60^\circ$ .

Answer

I) Tells us the perimeter of the triangle.

II) Tells us that FHT is an equilateral triangle.

Taking these statements together we are able to find the side length by dividing the perimeter from statement I, by 3 since all side lengths of an equilateral are the same by statement II.

$s=\frac{P}{3} \rightarrow s=\frac{114}{3}=38ft$

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Question

Given equilateral triangles $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , which, if either, is longer, $\overline{AB}$ or $\overline{DE}$ ?

Statement 1:

Statement 2: $BC \cdot EF = 50$

Answer

All sides of an equilateral triangle have the same measure, so we can let be the common sidelength of $\bigtriangleup ABC$ , and be that of $\bigtriangleup DEF$ .

Statement 1 can be rewritten as ; Statement 2 can be rewritten as . The equivalent question is whether we can determine which, if either, is greater, or . The two statements together are insufficient to answer the question, however; 5 and 10 have sum 15 and product 50, but we cannot determine without further information whether and , or vice versa. Therefore, we do not know for sure whether a side of $\bigtriangleup ABC$ is longer than a side of $\bigtriangleup DEF$ - specifically, which of $\overline{AB}$ or $\overline{DE}$ is longer.

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Question

Given two equilateral triangles $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , which, if either, is greater, or ?

Statement 1:

Statement 2:

Answer

An equilateral triangle has three sides of equal length, so and .

Assume Statement 1 alone. Since , then, by substitution, .

Assume Statement 2 alone. Since , it follows that , and again by substitution, .

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Question

What is the length of side $\overline{BC}$ of equilateral triangle $\bigtriangleup ABC$ ?

Statement 1: , , and are all located on a circle with area $576 \pi$ .

Statement 2: The midpoints of all three sides are located on a circle with circumference $24 \pi$ .

Answer

We demonstrate that either statement alone yields sufficient information by noting that the circle that includes all three vertices of a triangle - described in Statement 1 - is its circumscribed circle, and that the circle that includes all three midpoints of the sides of an equilateral triangle - described in Statement 2 - is its inscribed circle. We examine this figure below, which shows the triangle, both circles, and the three altitudes:

The three altitudes intersect at , which divides each altitude into two segments whose lengths have ratio 2:1. is the center of both the circumscribed circle, whose radius is , and the inscribed circle, whose radius is .

Therefore, from Statement 1 alone and the area formula for a circle, we can find from the area $576\pi$ of the circumscribed circle:

$\pi (OC)^{2} =a$

$\pi (OC)^{2} = 576 \pi$

$(OC)^{2} =576$

From Statement 2 alone and the circumference formula for a cicle, we can find from the circumference $24\pi$ of the inscribed circle:

$2 \pi \cdot OM = c$

$2 \pi \cdot OM = 24 \pi$

By symmetry, $\bigtriangleup COM$ is a 30-60-90 triangle, and either way, $CM = 12\sqrt{3}$ , and $BC =2 \cdot CM =2 \cdot 12\sqrt{3}= 24 \sqrt{3}$ .

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Question

What is the length of side $\overline{AB}$ of equilateral triangle $\bigtriangleup ABC$ ?

Statement 1: $\overline{BC}$ is a diagonal of Rectangle with area 30.

Statement 2: $\overline{AC}$ is a diagonal of Square with area 36.

Answer

An equilateral triangle has three sides of equal measure, so if the length of any one of the three sides can be determined, the lengths of all three can be as well.

Assume Statement 1 alone. $\overline{BC}$ is a diagonal of a rectangle of area 30. However, neither the length nor the width can be determined, so the length of this segment cannot be determined with certainty.

Assume Statement 2 alone. A square with area 36 has sidelength the square root of this, or 6; its diagonal, which is $\overline{AC}$ , has length $\sqrt{2}$ times this, or $6 \sqrt{2}$ . This is also the length of $\overline{AB}$ .

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Question

You are given two equilateral triangles $\bigtriangleup ABC$ and $\bigtriangleup DEF$ .

Which, if either, is greater, or ?

Statement 1: The perimeters of $\bigtriangleup DEF$ and $\bigtriangleup ABC$ are equal.

Statement 2: The areas of $\bigtriangleup DEF$ and $\bigtriangleup ABC$ are equal.

Answer

Assume Statement 1 alone, and let be the common perimeter of the triangles. Since an equilateral triangle has three sides of equal length, $AB = \frac{1}{3} P$ and $DE = \frac{1}{3} P$ , so .

Assume Statement 2 alone, and let be the common area of the triangles. Using the area formula for an equilateral triangle, we can note that:

$A = \frac{s^{2}\sqrt{3}}{4}$

$A = \frac{ (AB)^{2}\sqrt{3}}{4}$ and $A = \frac{ (DE)^{2}\sqrt{3}}{4}$ ,

so

$\frac{ (AB)^{2}\sqrt{3}}{4}= \frac{ (DE)^{2}\sqrt{3}}{4}$

$\frac{ (AB)^{2}\sqrt{3}}{4}\cdot \frac{4}{\sqrt{3}}= \frac{ (DE)^{2}\sqrt{3}}{4} \cdot \frac{4}{\sqrt{3}}$

$(AB)^{2} = (DE)^{2}$

.

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Question

Given equilateral triangles $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , which, if either, is longer, $\overline{AB}$ or $\overline{DE}$ ?

Statement 1:

Statement 2: $AC \cdot DF = 144$

Answer

All sides of an equilateral triangle have the same measure, so we can let be the common sidelength of $\bigtriangleup ABC$ , and be that of $\bigtriangleup DEF$ .

Statement 1 can be rewritten as ; Statement 2 can be rewritten as . The equivalent question is whether we can determine which, if either, is greater, or .

Statement 1 alone yields insufficient information; for example, the two numbers added together could be 10 and 14, but it is impossible to determine whether or is the greater of the two. Statement 2 alone is also insufficient, for a similar reason; for example, the two numbers could be 9 and 16, but again, either or could be the greater.

Now assume both statements. The only two numbers that can be added to yield a sum of 24 and multiplied to yield a product of 144 are 12 and 12; therefore, , and $\bigtriangleup ABC$ and $\bigtriangleup DEF$ have the same sidelengths. Specifically, $\overline{AB}$ and $\overline{DE}$ have the same length.

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Question

$\bigtriangleup ABC$ is equilateral. $\bigtriangleup DEF$ may or may not be equilateral.

which, if either, is longer, $\overline{AB}$ or $\overline{DE}$ ?

Statement 1:

Statement 2: and

Answer

Assume Statement 1 alone. $\bigtriangleup ABC$ is equilateral, so . Also, by the Triangle Inequality, the sum of the lengths of two sides of a triangle must exceed the third, so . From Statement 1, , so by substitution, , and .

Statement 2 alone provides insufficient information. For example, assume $\bigtriangleup ABC$ is an equilateral triangle with sidelength 9. If $\bigtriangleup DEF$ is an equilateral triangle with sidelength 8, the conditions of the statement hold, and . However, if $\bigtriangleup DEF$ is a right triangle in which , , and , the conditions of the statement still hold, but .

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Question

Given equilateral triangle $\bigtriangleup ABC$ and right triangle $\bigtriangleup DEF$ , which, if either, is longer, $\overline{AB}$ or $\overline{DE}$ ?

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

Statement 2: $\overline{AC} \cong \overline{DF}$

Answer

Assume Statement 1 alone. Since all three sides of $\bigtriangleup ABC$ are congruent - specifically, $\overline{AB} \cong \overline{BC}$ - and $\overline{BC} \cong \overline{EF}$ , it follows by transitivity that $\overline{AB} \cong \overline{EF}$ . However, no information is given as to whether $\overline{DE}$ has length greater than, equal to, or less than $\overline{EF}$ , so it cannot be determined which of $\overline{AB}$ and $\overline{DE}$ , if either, is the longer. By a similar argument, Statement 2 yields insufficient information.

Now assume both statements are true. $\overline{DF}$ and $\overline{EF}$ are each congruent to one of the congruent sides of equilateral $\bigtriangleup ABC$ and are therefore congruent to each other. However, the hypotenuse of a right triangle must be longer than both legs, so the hypotenuse of $\bigtriangleup DEF$ is $\overline{DE}$ . $\overline{DE}$ is also longer than any segment congruent to one of the legs, which includes all three sides of $\bigtriangleup ABC$ - specificially, $\overline{DE}$ is longer than $\overline{AB}$ .

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Question

Given equilateral triangle $\bigtriangleup ABC$ and right triangle $\bigtriangleup DEF$ , which, if either, is longer, $\overline{AB}$ or $\overline{DE}$ ?

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

Statement 2: $\angle D$ is a right angle.

Answer

Assume Statement 1 alone. Since all three sides of $\bigtriangleup ABC$ are congruent - specifically, $\overline{AB} \cong \overline{BC}$ - and $\overline{BC} \cong \overline{EF}$ , it follows by transitivity that $\overline{AB} \cong \overline{EF}$ . However, no information is given as to whether $\overline{DE}$ has length greater then, equal to, or less than $\overline{EF}$ , so which of $\overline{AB}$ and $\overline{DE}$ , if either, is the longer cannot be answered.

Assume Statement 2 alone. Since $\angle D$ is the right angle of $\bigtriangleup DEF$ , $\overline{EF}$ is the hypotenuse and this the longest side, so and . However, no comparisons with the sides of $\bigtriangleup ABC$ can be made.

Now assume both statements are true. as a consequence of Statement 1, and as a consequence of Statement 2, so .

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