GRE Triangle Questions Practice Questions with Answers

Concept Explanation

GRE triangle questions focus on the properties, area, and side relationships of three-sided polygons within the Quantitative Reasoning section. To solve these effectively, you must understand a few fundamental rules that govern all triangles. First, the sum of the interior angles in any triangle is always $180^\circ$. Second, the Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This is a common trap on the exam. Third, the area is calculated using the formula $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$, where the height is the perpendicular distance from the base to the opposite vertex.

The GRE often tests specific types of triangles. Isosceles triangles have two equal sides and two equal opposite angles, while equilateral triangles have three equal sides and three $60^\circ$ angles. Right triangles are particularly important; they follow the Pythagorean theorem: $a^2 + b^2 = c^2$. You should memorize common Pythagorean triples like 3-4-5 and 5-12-13, as well as the ratios for special right triangles: the 45-45-90 triangle (sides $x, x, x\sqrt{2}$) and the 30-60-90 triangle (sides $x, x\sqrt{3}, 2x$). For deeper study into quantitative patterns, you might use an AI Question Generator to produce variations of these geometry problems.

Solved Examples

1. Problem: In a right triangle, one leg has a length of 8 and the hypotenuse has a length of 17. What is the length of the other leg?
   1. Identify the knowns: $a = 8$, $c = 17$.
   2. Apply the Pythagorean theorem: $8^2 + b^2 = 17^2$.
   3. Calculate the squares: $64 + b^2 = 289$.
   4. Subtract 64 from both sides: $b^2 = 225$.
   5. Take the square root: $b = 15$. The length of the other leg is 15.
2. Problem: An isosceles triangle has two sides of length 10 and an included angle of $120^\circ$. What is the area of the triangle?
   1. Split the isosceles triangle into two congruent right triangles by drawing an altitude from the vertex of the $120^\circ$ angle.
   2. The altitude bisects the $120^\circ$ angle into two $60^\circ$ angles, creating two 30-60-90 triangles.
   3. In a 30-60-90 triangle, the hypotenuse is 10. The side opposite the $30^\circ$ angle (the altitude) is $\frac{10}{2} = 5$.
   4. The side opposite the $60^\circ$ angle (half the base) is $5\sqrt{3}$. Thus, the full base is $10\sqrt{3}$.
   5. Calculate area: $\text{Area} = \frac{1}{2} \times 10\sqrt{3} \times 5 = 25\sqrt{3}$.
3. Problem: A triangle has sides of length 4 and 11. What is the range of possible lengths for the third side, $x$?
   1. Use the Triangle Inequality Theorem: The sum of any two sides must be greater than the third.
   2. Condition 1: $4 + 11 > x \rightarrow 15 > x$.
   3. Condition 2: $4 + x > 11 \rightarrow x > 7$.
   4. The range is $7 < x < 15$.

Practice Questions

1. A right triangle has legs of length 5 and 12. What is the perimeter of the triangle?
2. In triangle ABC, the measure of angle A is $50^\circ$ and the measure of angle B is $70^\circ$. What is the measure of the exterior angle at vertex C?
3. An equilateral triangle has a side length of 6. What is its area?

4. If the sides of a triangle are in the ratio 3:4:5 and the perimeter is 48, what is the length of the longest side?
5. Quantity A: The area of a triangle with base 10 and height 8. Quantity B: The area of a circle with radius 4. (Compare the two quantities).
6. A 30-60-90 triangle has a hypotenuse of length 10. What is the area of this triangle?
7. A triangle has side lengths of 7, 24, and 25. Is this a right triangle?
8. In an isosceles triangle, the vertex angle is $40^\circ$. What is the measure of one of the base angles?
9. Can a triangle have side lengths of 3, 8, and 12?
10. A square is inscribed in a right isosceles triangle such that one side of the square lies on the hypotenuse. If the legs of the triangle are $\sqrt{2}$, what is the area of the square?

Answers & Explanations

1. 30: First, find the hypotenuse using $5^2 + 12^2 = c^2$, so $25 + 144 = 169$, meaning $c = 13$. Perimeter = $5 + 12 + 13 = 30$.
2. 120: The sum of angles in a triangle is $180^\circ$. Angle C = $180 - (50 + 70) = 60^\circ$. The exterior angle is supplementary to the interior angle: $180 - 60 = 120^\circ$.
3. $9\sqrt{3}$: The area of an equilateral triangle is $\frac{s^2\sqrt{3}}{4}$. Plugging in 6: $\frac{36\sqrt{3}}{4} = 9\sqrt{3}$.
4. 20: Let the sides be $3x, 4x, \text{ and } 5x$. $3x + 4x + 5x = 48 \rightarrow 12x = 48 \rightarrow x = 4$. Longest side = $5(4) = 20$.
5. Quantity B is greater: Quantity A = $\frac{1}{2} \times 10 \times 8 = 40$. Quantity B = $\pi \times 4^2 = 16\pi \approx 16 \times 3.14 = 50.24$.
6. $\frac{25\sqrt{3}}{2}$: In a 30-60-90 triangle with hypotenuse 10, the short leg is 5 and the long leg is $5\sqrt{3}$. Area = $\frac{1}{2} \times 5 \times 5\sqrt{3} = 12.5\sqrt{3}$.
7. Yes: Check if $7^2 + 24^2 = 25^2$. $49 + 576 = 625$, and $25^2 = 625$. Since the equation holds, it is a right triangle.
8. $70^\circ$: The sum is $180^\circ$. Let base angles be $x$. $40 + 2x = 180 \rightarrow 2x = 140 \rightarrow x = 70$.
9. No: By the Triangle Inequality Theorem, $3 + 8$ must be greater than 12. Since $11 < 12$, these side lengths cannot form a triangle.
10. $\frac{4}{9}$: This requires geometry visualization. In a right isosceles triangle with legs $\sqrt{2}$, the hypotenuse is 2. The square's side $s$ creates smaller triangles. Using ratios, the side of the square is $\frac{2}{3}$, so area is $\frac{4}{9}$. For more complex geometry, check GRE Prep resources.

1. If a triangle has sides of 5 and 10, which of the following could be the third side?

• A4
• B5
• C8
• D15

Frequently Asked Questions

What is the most important triangle rule for the GRE?

The most vital rule is that the sum of interior angles always equals $180^\circ$, which allows you to find missing angles in almost any problem. Additionally, the Pythagorean theorem for right triangles is frequently tested.

How do I identify a 30-60-90 triangle on the test?

You can identify it if one angle is $90^\circ$ and another is $30^\circ$ or $60^\circ$, or if the side lengths follow the $x : x\sqrt{3} : 2x$ ratio. These triangles often appear in problems involving equilateral triangles split in half.

Can the GRE ask about the Law of Sines or Cosines?

No, the GRE Quantitative Reasoning section does not require advanced trigonometry like the Law of Sines or Law of Cosines. You only need to know basic properties of right triangles and special right triangle ratios.

What is the Triangle Inequality Theorem?

This theorem states that the length of any side of a triangle must be strictly less than the sum of the other two sides and strictly greater than the difference of the other two sides. It is a common concept used in Quantitative Comparison questions.

How is the height of a triangle defined?

The height, or altitude, is the perpendicular line segment from a vertex to the line containing the opposite side. In a right triangle, the two legs can serve as the base and the height for area calculations.