GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #1 : Calculating The Height Of An Equilateral Triangle

If the area of an equilateral is \(\displaystyle 21\), given a base of \(\displaystyle 6\), what is the height of the triangle?

Possible Answers:

\(\displaystyle 7\)

\(\displaystyle 42\)

\(\displaystyle 10\)

\(\displaystyle 14\)

Correct answer:

\(\displaystyle 14\)

Explanation:

We derive the height formula from the area of the triangle formula:

\(\displaystyle A=\frac{(bh)}2{}\)

\(\displaystyle h=\frac{2A}{b}\)

\(\displaystyle =\frac{(2*21)}{3}\)

\(\displaystyle =14\)

Example Question #2 : Calculating The Height Of An Equilateral Triangle

What is the height of an equilateral triangle with sidelength 20?

Possible Answers:

\(\displaystyle 5 \sqrt{3}\)

\(\displaystyle 10\)

\(\displaystyle 10 \sqrt{3}\)

\(\displaystyle 10 \sqrt{2}\)

\(\displaystyle \frac{10 \sqrt{2}}{3}\)

Correct answer:

\(\displaystyle 10 \sqrt{3}\)

Explanation:

The area of an equilateral triangle with sidelength \(\displaystyle s = 20\) is 

\(\displaystyle A = \frac{s^{2}\sqrt{3}}{4} = \frac{20^{2}\sqrt{3}}{4} = \frac{400\sqrt{3}}{4} = 100\sqrt{3}\)

Using this area for \(\displaystyle A\) and 20 for \(\displaystyle b\) in the general triangle formula, we can obtain \(\displaystyle h\):

\(\displaystyle \frac{1}{2} bh = A\)

\(\displaystyle \frac{1}{2}\cdot 20 \cdot h =100\sqrt{3}\)

\(\displaystyle 10 \cdot h =100\sqrt{3}\)

\(\displaystyle 10 \cdot h \div 10 =100\sqrt{3}\div 10\)

\(\displaystyle h=10\sqrt{3}\)

Example Question #551 : Problem Solving Questions

An equilateral triangle has a side length of \(\displaystyle 5\). What is the height of the triangle?

Possible Answers:

\(\displaystyle \frac{2\sqrt{5}}{3}\)

\(\displaystyle \frac{5\sqrt{3}}{2}\)

\(\displaystyle 5\sqrt{2}\)

\(\displaystyle \frac{3\sqrt{2}}{5}\)

\(\displaystyle 10\)

Correct answer:

\(\displaystyle \frac{5\sqrt{3}}{2}\)

Explanation:

The height of an upright equilateral triangle is the perpendicular distance from the center of its base to its top. We can imagine that this line cuts the equilateral triangle into two congruent right triangles whose height is half the length of the original base and whose hypotenuse is the original side length. In these two congruent triangles, their base, which is the height of the equilateral triangle, is the only unknown side length, so we can use the Pythagorean theorem to solve for it:

\(\displaystyle a^2+b^2=c^2\)

\(\displaystyle (\frac{5}{2})^2+b^2=5^2\)

\(\displaystyle b^2=5^2-(\frac{5}{2})^2=\frac{75}{4}\)

\(\displaystyle b=\sqrt{\frac{75}{4}}=\frac{\sqrt{3*25}}{2}=\frac{5\sqrt{3}}{2}\)

Example Question #1 : Calculating The Height Of An Equilateral Triangle

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\(\displaystyle ABC\) is an equilateral triangle, with a side length of \(\displaystyle 3\). What is the height of the triangle?

Possible Answers:

\(\displaystyle 3\sqrt{3}\)

\(\displaystyle 2\sqrt{3}\)

\(\displaystyle \frac{3\sqrt{3}}{2}\)

\(\displaystyle \frac{\sqrt{3}}{2}\)

\(\displaystyle \frac{3}{2}\)

Correct answer:

\(\displaystyle \frac{3\sqrt{3}}{2}\)

Explanation:

We know the length of the side, therefore we can use the formula for the height in an equilateral triangle:

 \(\displaystyle h=s\frac{\sqrt{3}}{2}\), where \(\displaystyle s\) is the length of a side and \(\displaystyle h\) the length of the height.

Therefore, the final answer is \(\displaystyle \frac{3\sqrt{3}}{2}\).

Example Question #1 : Calculating The Height Of An Equilateral Triangle

Given that an equilateral triangle has side lengths equal to \(\displaystyle 6\), determine it's height in simplest form.

Possible Answers:

\(\displaystyle \sqrt{28}\)

\(\displaystyle 9\)

\(\displaystyle \sqrt6\)

\(\displaystyle \sqrt{27}\)

\(\displaystyle 3\sqrt3\)

Correct answer:

\(\displaystyle 3\sqrt3\)

Explanation:

To solve, we must use pythagorean's theorem given that we know the hypotenuse is \(\displaystyle 6\) and one side length is \(\displaystyle 3\) \(\displaystyle \left (\frac{1}{2} \textup{ of } 6 \right )\). Therefore:

\(\displaystyle c^2=a^2+b^2\Rightarrow 6^2=a^2+3^2\Rightarrow 27=a^2\)

\(\displaystyle a=\sqrt{27}=\sqrt{3^2*3}=3\sqrt3\)

Example Question #551 : Gmat Quantitative Reasoning

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The area of an equilateral triangle \(\displaystyle ABC\) is \(\displaystyle 7\). What is the perimeter of \(\displaystyle ABC\)?

Possible Answers:

\(\displaystyle 3\sqrt{\frac{28}{\sqrt{3}}}\)

\(\displaystyle 3\sqrt{3}\)

\(\displaystyle 3\sqrt{\frac{16}{\sqrt{3}}}\)

\(\displaystyle 3\)

\(\displaystyle \frac{28}{\sqrt{3}}\)

Correct answer:

\(\displaystyle 3\sqrt{\frac{28}{\sqrt{3}}}\)

Explanation:

The area is given, which will allow us to calculate the side of the triangle and hence we can also find the perimeter.

The area for an equilateral triangle is given by the formula 

\(\displaystyle \frac{s^{2}\sqrt{3}}{4}\), where \(\displaystyle s\) is the length of the side of the triangle.

Therefore, \(\displaystyle s=\sqrt{\frac{4a}{\sqrt{3}}}\), where \(\displaystyle a\) is the area.

Thus \(\displaystyle s=\sqrt{\frac{28}{\sqrt{3}}}\), and the perimeter of an equilateral triangle is three times the side, hence, the final answer is \(\displaystyle 3\sqrt{\frac{28}{\sqrt{3}}}\).

Example Question #552 : Gmat Quantitative Reasoning

A given equilateral triangle has a side length of \(\displaystyle 9cm\). What is the perimeter of the triangle?

Possible Answers:

\(\displaystyle 36cm\)

\(\displaystyle 27cm\)

Not enough information provided.

\(\displaystyle 45cm\)

\(\displaystyle 18cm\)

Correct answer:

\(\displaystyle 27cm\)

Explanation:

An equilateral triangle with a side length \(\displaystyle x\) has a perimeter \(\displaystyle P=3x\).

Given: 

\(\displaystyle x=9cm\)

\(\displaystyle P=3(9cm)=27cm\)

Example Question #3 : Calculating The Perimeter Of An Equilateral Triangle

A given equilateral triangle has a side length of \(\displaystyle 17cm\). What is the perimeter of the triangle?

Possible Answers:

\(\displaystyle 51cm\)

\(\displaystyle 8.5cm\)

\(\displaystyle 85cm\)

\(\displaystyle 68cm\)

\(\displaystyle 34cm\)

Correct answer:

\(\displaystyle 51cm\)

Explanation:

An equilateral triangle with a side length \(\displaystyle x\) has a perimeter \(\displaystyle P=3x\).

Given: 

\(\displaystyle x=17cm\)

\(\displaystyle P=3(17cm)=51cm\)

Example Question #4 : Calculating The Perimeter Of An Equilateral Triangle

A given equilateral triangle has a side length of \(\displaystyle 21cm\). What is the perimeter of the triangle?

Possible Answers:

\(\displaystyle 42cm\)

Not enough information provided.

\(\displaystyle 84cm\)

\(\displaystyle 63cm\)

\(\displaystyle 10.5cm\)

Correct answer:

\(\displaystyle 63cm\)

Explanation:

An equilateral triangle with a side length \(\displaystyle x\) has a perimeter \(\displaystyle P=3x\).

Given: 

\(\displaystyle x=21cm\)

\(\displaystyle P=3(21cm)=63cm\)

Example Question #3 : Calculating The Perimeter Of An Equilateral Triangle

Given the following regarding Triangle \(\displaystyle HFT\).

I) \(\displaystyle \small \angle H=\angle F=60^{\circ}\)

II) Side \(\displaystyle t\) is equal to \(\displaystyle 7\) light-years

What is the length of side \(\displaystyle f\)?

Possible Answers:

\(\displaystyle \sqrt{7}\) light years

\(\displaystyle 7\) light years

\(\displaystyle 21\) light years

\(\displaystyle 49\) light years

\(\displaystyle 14\) light years

Correct answer:

\(\displaystyle 7\) light years

Explanation:

The trick to this one is to carefully put together what you are given.

We know that two of our angles are equal to 60 degrees. This means that the last angle is also 60 degrees. This make HFT an equilateral triangle.

Equilateral triangles always have equal sides and equal angles, so our last side has to be 7 light years as well. 

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