GED Math: Triangles and the Pythagorean Relationship
Some right triangles have fixed side ratios worth memorizing. In a 45°–45°–90° triangle (an isosceles right triangle), the two legs are equal and leg : leg : hypotenuse = x : x : x√2. In a 30°–60°–90° triangle, short leg : long leg : hypotenuse = x : x√3 : 2x, where the short leg is opposite the 30° angle. Also memorize the Pythagorean triples 3-4-5, 5-12-13, 8-15-17, and 7-24-25 and their multiples: any triangle with those side lengths is a right triangle. To test whether three sides form a right triangle, check whether the two smaller squares add up to the largest square.
1. Naming Triangles and the 180° Rule
A triangle is a closed, three-sided plane figure. It has three sides, three vertices (corners), and three interior angles. A triangle is named by listing its vertices in any order, with the symbol Δ: ΔDEF has sides DE, EF, and DF. The most important fact about triangles is that the three interior angles always add to 180°. If you know two angles, subtract their sum from 180° to find the third. When angles are given as expressions, add them, set the total equal to 180, and solve the equation. Test tip: classify a triangle by its properties (side marks and angle measures), not by how it looks in the drawing.
Worked Example: In ΔABC, ∠A = 55° and ∠B = 100°. Find ∠C.
55 + 100 + C = 180, so 155 + C = 180 and C = 25°.
1. What are the sides of ΔPQR?
Explanation: Each pair of vertices forms a side: PQ, QR, and PR. The letters P, Q, R alone are the vertices.
2. What is the sum of the interior angles of any triangle?
Explanation: The three angles of every triangle add to 180°.
3. In ΔABC, ∠A = 55° and ∠B = 100°. What is ∠C?
Explanation: 180 − (55 + 100) = 25.
4. One angle of a scalene triangle is 38° and another is 56°. What is the third angle?
Explanation: 38 + 56 = 94, and 180 − 94 = 86.
5. A triangle has angles of 72° and 44°. What is the third angle?
Explanation: 72 + 44 = 116, and 180 − 116 = 64.
6. A right triangle has one angle of 28° (besides the right angle). What is the remaining angle?
Explanation: The right angle is 90°, so the other two add to 90: 90 − 28 = 62.
7. The angles of a triangle are x, 2x, and 3x. What is the largest angle?
Explanation: 6x = 180, so x = 30. The largest angle is 3(30) = 90.
8. The angles of a triangle are (x + 10)°, 2x°, and (x + 30)°. What is the smallest angle?
Explanation: 4x + 40 = 180, so x = 35. The angles are 45°, 70°, and 65°; the smallest is 45°.
2. Classifying Triangles by Their Sides
Triangles are classified by side length. An equilateral triangle has three equal sides and therefore three equal angles of 60°. An isosceles triangle has exactly two equal sides, and the two angles opposite those sides are equal. A scalene triangle has no equal sides and no equal angles. In drawings, sides with the same number of tick marks are equal. The perimeter of a triangle is the sum of its three sides. In an isosceles triangle, the unequal angle is the vertex angle and the two equal angles are the base angles.
Worked Example: An isosceles triangle has a vertex angle of 40°. Find each base angle.
The base angles are equal, so 2b + 40 = 180, 2b = 140, b = 70°.
1. A triangle has sides 5, 5, and 5. What kind of triangle is it?
Explanation: All three sides are equal: equilateral.
2. A triangle has sides 7, 7, and 10. What kind of triangle is it?
Explanation: Exactly two sides are equal: isosceles.
3. A triangle has sides 6, 8, and 10. What kind of triangle is it by its sides?
Explanation: No two sides are equal: scalene.
4. What is each angle of an equilateral triangle?
Explanation: The three equal angles share 180°: 180 ÷ 3 = 60.
5. Which triangle has no equal sides and no equal angles?
Explanation: A scalene triangle has all different sides and all different angles.
6. An isosceles triangle has a vertex angle of 40°. What is each base angle?
Explanation: (180 − 40) ÷ 2 = 70.
7. An isosceles triangle has one base angle of 50°. What is the vertex angle?
Explanation: The two base angles are 50° each (100° total), so the vertex angle is 180 − 100 = 80.
8. The perimeter of an equilateral triangle is 36 cm. How long is each side?
Explanation: 36 ÷ 3 = 12.
3. Classifying Triangles by Their Angles
Triangles are also classified by their largest angle. A right triangle has one 90° angle. An acute triangle has all angles less than 90°. An obtuse triangle has one angle greater than 90°. A triangle can have at most one right angle or one obtuse angle, because the angle total is only 180°. Every triangle can be named two ways, once by sides and once by angles, such as a right isosceles triangle or an obtuse scalene triangle. Test tip: the right angle may be at the top or at a corner you do not expect, so look for the square marker or check the angle measures.
Worked Example: What kind of triangle is ΔPQR with a right angle at P and sides PQ = PR = 10?
By sides: two sides are equal, so it is isosceles. By angles: P is a right angle, so it is a right triangle. It is a right isosceles triangle.
1. A triangle has angles 50°, 45°, and 85°. What kind of triangle is it by its angles?
Explanation: All angles are less than 90°: acute.
2. A triangle has an angle of 125°. What kind of triangle is it by its angles?
Explanation: One angle is greater than 90°: obtuse.
3. A triangle has angles 30°, 60°, and 90°. What is it called?
Explanation: It has a 90° angle, so it is right, and all three angles (and sides) are different, so it is scalene.
4. A triangle has angles 40°, 40°, and 100°. What is it called?
Explanation: The 100° angle makes it obtuse, and the two equal angles make it isosceles.
5. A triangle has angles 90°, 45°, and 45°. What is it called?
Explanation: It has a right angle and two equal angles.
6. Can a triangle have two right angles?
Explanation: Two right angles already total 180°, leaving 0° for the third angle.
7. A triangle has angles of 20°, 20°, and 140°. Which statement is true?
Explanation: 140° is greater than 90° (obtuse) and two angles are equal (isosceles).
8. In ΔABC, ∠A = 62° and ∠B = 28°. What kind of triangle is it?
Explanation: ∠C = 180 − 90 = 90°, so it is a right triangle.
4. The Pythagorean Relationship: Finding the Hypotenuse
In a right triangle, the side opposite the right angle is the hypotenuse and is always the longest side. The other two sides are the legs. The Pythagorean relationship says a2 + b2 = c2, where a and b are the legs and c is the hypotenuse. To find the hypotenuse: (1) square both legs, (2) add the squares, (3) take the square root of the sum. If the answer is not a whole number, use a calculator and round as directed. Test tip: watch for the whole-number triples 3-4-5 and 5-12-13 (and their multiples such as 6-8-10 and 9-12-15), which appear often.
Worked Example: The legs of a right triangle are 3 ft and 4 ft. Find the hypotenuse.
c2 = 32 + 42 = 9 + 16 = 25, so c = √25 = 5 ft.
1. In a right triangle, which side is the hypotenuse?
Explanation: The hypotenuse is opposite the right angle and is the longest side.
2. The legs of a right triangle are 3 ft and 4 ft. What is the hypotenuse?
Explanation: 32 + 42 = 25, and √25 = 5.
3. The legs of a right triangle are 9 yd and 12 yd. What is the hypotenuse?
Explanation: 81 + 144 = 225, and √225 = 15.
4. The legs of a right triangle are 1.5 cm and 2 cm. What is the hypotenuse?
Explanation: 2.25 + 4 = 6.25, and √6.25 = 2.5.
5. The two shorter sides of a right triangle are 18 ft and 24 ft. What is the third side?
Explanation: 324 + 576 = 900, and √900 = 30 (a 3-4-5 triangle times 6).
6. The legs of a right triangle are 7 cm and 10 cm. What is the hypotenuse to the nearest tenth?
Explanation: 49 + 100 = 149, and √149 ≈ 12.2.
7. The legs of a right triangle are 5 m and 12 m. What is the hypotenuse?
Explanation: 25 + 144 = 169, and √169 = 13.
8. Which equation correctly describes a right triangle with legs a and b and hypotenuse c?
Explanation: The squares of the legs add to the square of the hypotenuse.
5. Finding a Leg and Real-World Right Triangles
To find a leg, rearrange the same relationship: a2 = c2 − b2. Subtract the square of the known leg from the square of the hypotenuse, then take the square root. Use this for ladders, ramps, diagonals of rectangles, and distances on a coordinate grid. Steps: (1) sketch the right triangle and label what you know; (2) decide whether the unknown side is the hypotenuse or a leg; (3) substitute into a2 + b2 = c2; (4) solve and round if asked. To find the distance between two points on a coordinate plane, count the horizontal and vertical differences; these are the legs of a right triangle, and the distance is the hypotenuse.
Worked Example: A 15-ft ladder rests 4 ft from the base of a wall. How high up the wall does it reach, to the nearest tenth?
42 + b2 = 152, so 16 + b2 = 225, b2 = 209, and b = √209 ≈ 14.5 ft.
1. A 15-ft ladder rests 4 ft from the base of a wall. How high up the wall does it reach, to the nearest tenth of a foot?
Explanation: b2 = 152 − 42 = 225 − 16 = 209, so b ≈ 14.5.
2. The hypotenuse of a right triangle is 39 in. and one leg is 15 in. What is the other leg?
Explanation: 392 − 152 = 1521 − 225 = 1296, and √1296 = 36.
3. A right triangle has hypotenuse 10 mm and one leg 6 mm. What is the other leg?
Explanation: 100 − 36 = 64, and √64 = 8.
4. A right triangle has hypotenuse 30 in. and one leg 15 in. What is the other leg to the nearest tenth?
Explanation: 900 − 225 = 675, and √675 ≈ 26.0.
5. A rectangular frame is 14 ft long and 6 ft wide. How long is a diagonal brace, to the nearest tenth of a foot?
Explanation: The brace is the hypotenuse: 142 + 62 = 196 + 36 = 232, and √232 ≈ 15.2.
6. A ramp rises 3 ft over a horizontal distance of 12 ft. How long is the ramp surface, to the nearest tenth?
Explanation: 32 + 122 = 9 + 144 = 153, and √153 ≈ 12.4.
7. Points A (−2, 3) and C (3, −1) are on a coordinate plane. What is the distance from A to C, to the nearest tenth?
Explanation: The horizontal leg is 5 and the vertical leg is 4. 25 + 16 = 41, and √41 ≈ 6.4.
8. What is the distance between (−3, 4) and (5, −2)?
Explanation: The legs are 8 and 6, and √(64 + 36) = √100 = 10.
6. Special Right Triangles
Some right triangles have fixed side ratios worth memorizing. In a 45°–45°–90° triangle (an isosceles right triangle), the two legs are equal and leg : leg : hypotenuse = x : x : x√2. In a 30°–60°–90° triangle, short leg : long leg : hypotenuse = x : x√3 : 2x, where the short leg is opposite the 30° angle. Also memorize the Pythagorean triples 3-4-5, 5-12-13, 8-15-17, and 7-24-25 and their multiples: any triangle with those side lengths is a right triangle. To test whether three sides form a right triangle, check whether the two smaller squares add up to the largest square.
Worked Example: A 30°–60°–90° triangle has a hypotenuse of 12. Find the short leg and the long leg.
The hypotenuse is 2x = 12, so the short leg x = 6. The long leg is 6√3 ≈ 10.4.
1. A 45°–45°–90° triangle has legs of 7 cm. What is the hypotenuse, to the nearest tenth?
Explanation: The hypotenuse is 7√2 ≈ 9.9.
2. The hypotenuse of a 45°–45°–90° triangle is 10√2. How long is each leg?
Explanation: Hypotenuse = leg × √2, so the leg is 10.
3. The short leg of a 30°–60°–90° triangle is 5. What is the hypotenuse?
Explanation: The hypotenuse is twice the short leg: 2(5) = 10.
4. The short leg of a 30°–60°–90° triangle is 5. What is the long leg, to the nearest tenth?
Explanation: The long leg is 5√3 ≈ 8.7.
5. The hypotenuse of a 30°–60°–90° triangle is 12. What is the short leg?
Explanation: The short leg is half the hypotenuse: 6.
6. Which set of side lengths forms a right triangle?
Explanation: 72 + 242 = 49 + 576 = 625 = 252.
7. A right triangle has legs 10 and 24. What is the hypotenuse?
Explanation: 10-24-26 is a multiple (×2) of the 5-12-13 triple: 100 + 576 = 676 = 262.
8. A right triangle has legs 9 and 12. What is the hypotenuse?
Explanation: 9-12-15 is 3 times the 3-4-5 triple.
Common Mistakes to Avoid
- Using the wrong total: the angles of a triangle add to 180°, not 360° (that is for four-sided figures).
- Classifying a triangle by its appearance instead of its marks and angle measures.
- Adding the legs (a + b) instead of squaring: the hypotenuse is not the sum of the legs.
- Forgetting the final square root and stopping at c2.
- Using the Pythagorean relationship on a triangle that is not a right triangle.
- When finding a leg, adding the squares instead of subtracting: a2 = c2 − b2.
- Calling any side the hypotenuse. It is always opposite the right angle and is the longest side.
- Mixing up the special-triangle ratios: √2 goes with 45°–45°–90° and √3 goes with 30°–60°–90°.
Frequently Asked Questions
What do the angles of a triangle add up to?
180°.
How are triangles classified?
By sides (equilateral, isosceles, scalene) and by angles (acute, right, obtuse).
What is the Pythagorean relationship?
In a right triangle, a2 + b2 = c2, where a and b are the legs and c is the hypotenuse.
How do I find a missing leg?
Subtract the square of the known leg from the square of the hypotenuse, then take the square root.
What are the special right triangles?
The 45°–45°–90° triangle has sides x, x, x√2. The 30°–60°–90° triangle has sides x, x√3, 2x.
How do I find the distance between two points?
Count the horizontal and vertical differences, then use the Pythagorean relationship; the distance is the hypotenuse.
Where can I practice more problems like these?
The GED Math practice in the GED Math QBank gives you more practice on this topic, and you can browse every other topic in the full GED Math QBank.