Triangle Fundamentals
A triangle has three sides and three angles. Two key rules govern all triangles on the GMAT:
Special Right Triangles
Area, Perimeter & Height
Similar Triangles
Two triangles are similar if they have the same angles. Their sides are proportional. This is a critical GMAT concept for finding missing lengths.
10 Triangle Traps
1. Height ≠ Side
The height of a triangle is perpendicular to the base — in an obtuse triangle, it falls outside the triangle.
2. Pythagorean triples — check multiples
A 6-8-10 triangle is right, not just 3-4-5. Divide by GCD to identify the base triple.
3. Triangle inequality
Three sides of 3, 4, 8 CANNOT form a triangle: $3+4=7 < 8$.
4. Equilateral ≠ "largest area"
Equilateral maximizes area for given perimeter, but a larger scalene can have more area.
5. 30-60-90: which leg is which?
The side OPPOSITE 30° is the shortest. The side opposite 60° is $\sqrt{3}$ × shorter.
6. Similar triangles: area ratio is squared
If sides are in ratio 2:3, areas are in ratio 4:9 — not 2:3.
7. Exterior angle theorem
An exterior angle equals the sum of the two non-adjacent interior angles.
8. Isosceles: equal angles are the BASE angles
The two equal sides are opposite the two equal base angles.
9. Altitude in equilateral
Altitude of equilateral triangle with side $s$ = $\frac{\sqrt{3}}{2}s$.
10. Median vs altitude
A median goes to the midpoint of the opposite side. An altitude goes perpendicular to it. They are NOT the same.
10 GMAT Practice Questions
In a right triangle, the two legs measure 5 and 12. What is the length of the hypotenuse?
(B) 13. This is the 5-12-13 Pythagorean triple. $\sqrt{5^2+12^2} = \sqrt{25+144} = \sqrt{169} = 13$.
A triangle has angles in the ratio 1:2:3. What is the measure of the largest angle?
(C) 90°. Angles sum to 180°. Ratio 1:2:3 → parts are $x, 2x, 3x$. $6x = 180$ → $x=30$. Largest = $3(30) = 90°$.
An equilateral triangle has a perimeter of 36. What is its area?
(A) $18\sqrt{3}$. Side = $36/3 = 12$. Area = $\frac{\sqrt{3}}{4}(12)^2 = \frac{\sqrt{3}}{4}(144) = 36\sqrt{3}$. Wait — $\frac{144\sqrt{3}}{4} = 36\sqrt{3}$. Answer is (C). (C) $36\sqrt{3}$ is correct.
Two similar triangles have corresponding sides in the ratio 3:5. If the area of the smaller triangle is 27 square inches, what is the area of the larger triangle?
(B) 75. Ratio of areas = $(3:5)^2 = 9:25$. If smaller area = 27, then $\frac{27}{9} \times 25 = 75$.
In a 30-60-90 triangle, the hypotenuse is 10. What is the length of the shorter leg?
(B) 5. In a 30-60-90 triangle, the shorter leg = hypotenuse/2 = 10/2 = 5.
Can a triangle have sides of length 4, 7, and 12?
(C) No. The triangle inequality requires: sum of any two sides > third side. $4+7=11 < 12$. This fails, so no such triangle can exist.
In right triangle ABC with right angle at C, what is the value of $\sin A \times \cos B$?
(Note: $\sin A = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos B = \frac{\text{adjacent}}{\text{hypotenuse}}$)
(D) $\frac{a^2}{c^2}$. In right triangle with right angle at C: $\sin A = \frac{a}{c}$ (opposite to A / hyp). $\cos B = \frac{a}{c}$ (adjacent to B is side $a$, hyp is $c$). So $\sin A \times \cos B = \frac{a}{c} \times \frac{a}{c} = \frac{a^2}{c^2}$.
Is triangle XYZ a right triangle?
(1) The three sides of XYZ measure 8, 15, and 17.
(2) One angle of XYZ measures 90°.
(D) Each alone sufficient. (1): 8-15-17 is a Pythagorean triple ($8^2+15^2=64+225=289=17^2$). Sufficient. (2): If one angle = 90°, then by definition it's a right triangle. Sufficient. Both alone → answer (D).
Triangle ABC has vertices at A(0,0), B(6,0), and C(3,4). What is the area of triangle ABC?
(C) 12. Base = AB = 6 (along x-axis). Height = y-coordinate of C = 4. Area = $\frac{1}{2}(6)(4) = 12$.
In isosceles triangle PQR, PQ = PR = 10 and QR = 12. What is the area of the triangle?
(C) 48. Drop altitude from P to midpoint M of QR. QM = 6. Height = $\sqrt{10^2 - 6^2} = \sqrt{64} = 8$. Area = $\frac{1}{2}(12)(8) = 48$.
Lesson Summary — Key Takeaways
Angle sum = 180° always
Any interior angle sum deviation from 180° means you've made an arithmetic error.
Memorize the triples
3-4-5, 5-12-13, 8-15-17, 7-24-25 and their multiples cover 90% of right triangle GMAT problems.
Similar triangles: area scales as k²
If sides scale by k, areas scale by k². This is the most common similar-triangle trap.
Height is perpendicular to base
Not a side — draw the altitude before computing area.