4.2 Triangles & Right Triangle Geometry

4.2 Triangles & Right Triangle Geometry

Triangles are the single most tested geometric figure on the GRE Quantitative Reasoning section. A thorough understanding of basic triangle properties, the Triangle Inequality Theorem, area calculations, isosceles triangle rules, Pythagorean triples, and special right triangles ($30^\circ-60^\circ-90^\circ$ and $45^\circ-45^\circ-90^\circ$) will allow you to solve even the most complex GRE geometry questions.

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1. Fundamental Triangle Properties

1. Sum of Interior Angles: The sum of the interior angles of any triangle is always $180^\circ$. $\angle A + \angle B + \angle C = 180^\circ$
2. Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of its two non-adjacent (remote) interior angles. $\text{Exterior Angle } \angle D = \angle A + \angle B$
3. Side-Angle Relationship:
   • The longest side of a triangle is always opposite the largest interior angle.
   • The shortest side is always opposite the smallest interior angle.
   • If two angles are equal, the sides opposite those angles are equal (and vice-versa).

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2. The Triangle Inequality Theorem

The Triangle Inequality Theorem governs the relationship between the three side lengths of a triangle.

Theorem: The sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side.

For a triangle with side lengths $a$, $b$, and $c$: $a + b > c, \quad a + c > b, \quad b + c > a$

The Third Side Rule (Double Inequality)

If you are given two sides of a triangle with lengths $a$ and $b$ (where $a \ge b$), the third side $c$ must satisfy:

$a - b < c < a + b$

• Lower Bound: The third side must be strictly greater than the positive difference of the other two sides.
• Upper Bound: The third side must be strictly less than the sum of the other two sides.

Example: If a triangle has sides of length 7 and 12, the third side $c$ must satisfy $12 - 7 < c < 12 + 7$, so $5 < c < 19$. The third side cannot be 5, cannot be 19, and cannot be negative.

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3. Classification of Triangles

Triangles are classified either by side lengths or by interior angle measures:

By Side Lengths:

• Scalene Triangle: All three sides have different lengths; all three interior angles have different measures.
• Isosceles Triangle: At least two sides are equal in length. The angles opposite these equal sides (the base angles) are also equal.
• Equilateral Triangle: All three sides are equal in length ($a = b = c$). All three interior angles measure $60^\circ$.

By Angle Measures:

• Acute Triangle: All three interior angles are strictly less than $90^\circ$.
• Right Triangle: Has exactly one $90^\circ$ right angle. The side opposite the right angle is the hypotenuse (the longest side).
• Obtuse Triangle: Has exactly one interior angle greater than $90^\circ$.

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4. Triangle Perimeter and Area

• Perimeter: The sum of all three side lengths: $P = a + b + c$
• Area Formula: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2}bh$ Important: The height ($h$) must be the perpendicular distance from the chosen base to the opposite vertex. In an obtuse triangle, the height may fall outside the physical boundary of the triangle.

Special Formula: Area of an Equilateral Triangle

For an equilateral triangle with side length $s$:

$\text{Height } h = \frac{s\sqrt{3}}{2}$

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

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5. Isosceles Triangle Properties & Rules

An isosceles triangle has two equal sides ($a = b$) and two equal base angles opposite those sides.

• Perpendicular Bisector / Altitude Property: Dropping an altitude (height) from the vertex angle between the two equal sides to the unequal base bisects both the base and the vertex angle.
• This altitude splits the isosceles triangle into two congruent right triangles, making it straightforward to apply the Pythagorean Theorem to find missing heights or base lengths.

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6. The Pythagorean Theorem & Pythagorean Triples

For any right triangle with legs $a$ and $b$ and hypotenuse $c$:

$a^2 + b^2 = c^2$

Common Pythagorean Triples

A Pythagorean triple is a set of three positive integers $(a, b, c)$ that satisfy $a^2 + b^2 = c^2$. Memorizing common triples—and their integer multiples—saves critical time on the GRE:

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7. Special Right Triangles

The GRE frequently tests two specific right triangles derived from regular geometric figures. Memorizing their exact side length ratios allows you to avoid calculating square roots manually.

1. $45^\circ-45^\circ-90^\circ$ Right Triangle (Isosceles Right Triangle)

Created by drawing a diagonal across a square.

• Angle Ratios: $45^\circ : 45^\circ : 90^\circ$
• Side Length Ratio: $x : x : x\sqrt{2}$

$\text{Leg} = x, \quad \text{Hypotenuse} = x\sqrt{2}$

$\text{Leg} = \frac{\text{Hypotenuse}}{\sqrt{2}} = \frac{\text{Hypotenuse} \cdot \sqrt{2}}{2}$

2. $30^\circ-60^\circ-90^\circ$ Right Triangle

Created by dropping an altitude down the center of an equilateral triangle.

• Angle Ratios: $30^\circ : 60^\circ : 90^\circ$

• Side Length Ratio: $x : x\sqrt{3} : 2x$

• Short Leg (opposite $30^\circ$) $= x$

• Long Leg (opposite $60^\circ$) $= x\sqrt{3}$

• Hypotenuse (opposite $90^\circ$) $= 2x$

Special Right Triangle Summary Table

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8. Worked GRE Step-by-Step Examples

Example 1: Multi-Step Right Triangle Calculation

Problem: In triangle $ABC$, angle $B = 90^\circ$ and angle $A = 30^\circ$. If side $AC = 12$, find the area of triangle $ABC$.

Solution:

1. Identify the Triangle Type: Since angle $B = 90^\circ$ and angle $A = 30^\circ$, angle $C = 180^\circ - (90^\circ + 30^\circ) = 60^\circ$. Triangle $ABC$ is a $30^\circ-60^\circ-90^\circ$ special right triangle.
2. Identify the Hypotenuse: The side opposite the $90^\circ$ angle is $AC = 12$.
3. Apply the $30^\circ-60^\circ-90^\circ$ Side Ratios ($x : x\sqrt{3} : 2x$): $\text{Hypotenuse } 2x = 12 \implies x = 6$
   • Short leg (opposite $30^\circ$, side $BC$) $= x = 6$.
   • Long leg (opposite $60^\circ$, side $AB$) $= x\sqrt{3} = 6\sqrt{3}$.
4. Calculate Area: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times BC \times AB = \frac{1}{2} \times 6 \times 6\sqrt{3} = 18\sqrt{3}$

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Example 2: Triangle Inequality Range

Problem: Two sides of a triangle have lengths 8 and 15. Which of the following could be the perimeter of the triangle? Select all that apply: (A) 30, (B) 32, (C) 40, (D) 46.

Solution:

1. Find the Range for the Third Side ($c$): $15 - 8 < c < 15 + 8 \implies 7 < c < 23$
2. Find the Range for the Perimeter ($P$): $\text{Perimeter } P = 8 + 15 + c = 23 + c$
   • Minimum bound for perimeter: $23 + 7 = 30$ (exclusive, so $P > 30$).
   • Maximum bound for perimeter: $23 + 23 = 46$ (exclusive, so $P < 46$). $\text{Valid Perimeter Range}: 30 < P < 46$
3. Evaluate the Options:
   • 30 is not valid because $c$ must be strictly greater than 7 ($P > 30$).
   • 32 is valid ($30 < 32 < 46$).
   • 40 is valid ($30 < 40 < 46$).
   • 46 is not valid because $c$ must be strictly less than 23 ($P < 46$).
4. Result: The valid perimeters are 32 and 40.

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9. GRE Exam Strategies & Common Pitfalls

• Do Not Assume a Right Triangle: Never use $a^2 + b^2 = c^2$ unless the problem explicitly states that an angle is $90^\circ$ or displays a right-angle box symbol.
• Recognize Equilateral Splitting: Whenever a problem mentions an equilateral triangle, remember that dropping a height splits it into two $30^\circ-60^\circ-90^\circ$ triangles.
• Pythagorean Triple Multiples: Watch for scaled triples such as $1.5-2-2.5$ or $0.6-0.8-1.0$. Dividing all sides by a common factor reveals the underlying $3-4-5$ or $5-12-13$ ratio instantly.