On the test
Geometry and Trigonometry is about 15% of the Math section, around 2 to 4 questions in each 22-question module, spread through the module from easier to harder.
5 to 7 of the 44 questions (about 2 to 4 in each module). Spread through the module, easier first and harder at the end.
How I approached it
A small story from Aime
When I see a right triangle question, I think about a ladder against a wall. The wall stands straight up, the ground lies flat, and the ladder makes a slanted line between them. If I know how far the bottom is from the wall and how long the ladder is, I do not need to climb up and measure. The height is already decided. A ladder on the same spot always reaches the same height, and a steeper ladder always reaches higher. People who build houses and fences use this fact every day. The SAT asks the same thing. It gives you two sides of a right triangle, or one side and an angle, and asks for something missing. You use a fixed rule to get it, and you do not need to guess.
Right triangles are one of the most predictable topics on the SAT. A few rules and a few special triangles, and you can answer most of these quickly and with confidence.
What is inside this part
Using a² + b² = c² to find a missing side of a right triangle.
Finding the hypotenuse · Finding a leg · Pythagorean triples · Real-life ladder and distance problems
Using the fixed side ratios of 45-45-90 and 30-60-90 triangles.
Diagonal of a square · Height of an equilateral triangle
Using the ratios of sides to an acute angle to find a trig value or a side.
SOH-CAH-TOA · Building a triangle from a ratio · Finding perimeter or area from a ratio
Using the fact that sin of an angle equals cos of its complement.
sin A = cos B · sin x° = cos(90 − x)°
How to know a question is from this part
A right triangle, a ladder, a diagonal, or a ratio like sin, cos or tan is mentioned.
- 'Right triangle' or 'right angle at B'
- sin, cos or tan with a fraction or decimal
- Sides like 3, 4, 5 or 5, 12, 13
- A ladder against a wall, a diagonal of a rectangle or square
- 'sin A = cos B' type statements
The lesson
1. What a right triangle is
A right triangle has one angle of exactly 90°. The two sides that make the right angle are called legs. The longest side, opposite the right angle, is called the hypotenuse.
The other two angles are acute (less than 90°), and together they add up to 90°, because all three must make 180°.
2. The Pythagorean theorem
In every right triangle, a² + b² = c², where a and b are the legs and c is the hypotenuse. It only works for right triangles, and c must be the longest side.
If you need the hypotenuse, add the squares of the legs and take the square root. If you need a leg, subtract the square of the known leg from the square of the hypotenuse, then take the square root.
A right triangle has legs 6 and 8. How long is the hypotenuse?
Show the steps
- c² = 6² + 8² = 36 + 64 = 100.
- c = √100 = 10.
10
3. Pythagorean triples
Some whole-number triangles come up again and again. Learn them and you save time: 3-4-5, 5-12-13, 8-15-17, 7-24-25. Any multiple also works: 6-8-10, 9-12-15, 10-24-26 (double 5-12-13) and so on.
If you spot a triple, you can write the third side without any calculating. A right triangle with a leg of 10 and a hypotenuse of 26 is 10-24-26, so the other leg is 24.
4. Special right triangles
The 45-45-90 triangle is half of a square. Its two legs are equal, and the hypotenuse is leg × √2. So a square with side 12 has diagonal 12√2.
The 30-60-90 triangle is half of an equilateral triangle. If the short leg (opposite 30°) is x, then the long leg (opposite 60°) is x√3 and the hypotenuse is 2x.
Bluebook shows these on the reference sheet, but you must know which side is which.
A square has a diagonal of 12√2. What is its area?
Show the steps
- The diagonal of a square is side × √2, so the side is 12.
- Area = 12 × 12 = 144.
144
5. Sine, cosine and tangent
Trigonometry names the ratios of sides. Pick one acute angle, called θ. The side across from it is the opposite. The side next to it (not the hypotenuse) is the adjacent. The hypotenuse is always the hypotenuse.
sin θ = opposite ÷ hypotenuse. cos θ = adjacent ÷ hypotenuse. tan θ = opposite ÷ adjacent. A memory aid is SOH-CAH-TOA.
The opposite and adjacent switch when you move to the other angle, but the hypotenuse never changes.
A right triangle has legs 5 and 12 and hypotenuse 13. For the angle opposite the side 5, what is the sine?
Show the steps
- Opposite side = 5, hypotenuse = 13.
- sin = opposite ÷ hypotenuse = 5/13.
5/13
6. Complementary angles: sin and cos
In a right triangle, the two acute angles add to 90°. The side opposite one angle is the side adjacent to the other. So sin of one angle equals cos of the other: sin A = cos B when A + B = 90°. Also, sin x° = cos(90 − x)°.
This idea appears on the SAT often, in a plain statement like 'sin A = 0.6, what is cos B?' with no figure needed.
7. Using the ratios to find sides
If you know a ratio and one side, you can build the triangle. Say tan A = 3/4. Then opposite : adjacent = 3 : 4. Because 3-4-5 is a triple, the hypotenuse is 5 for those sizes. Scale up to match the side you are given.
Example: tan A = 3/4 and hypotenuse = 20. Hypotenuse 5 became 20, so multiply by 4. The sides are 12, 16 and 20.
In a right triangle, angle A has tan A = 3/4 and the hypotenuse is 20. How long is the side opposite A?
Show the steps
- A 3-4-5 triangle scaled by 4 has sides 12, 16, 20.
- The side opposite A is the 3 part, so 3 × 4 = 12.
12
8. Traps and habits
Trap 1: using the Pythagorean theorem with a leg as c. The hypotenuse is always the longest. Trap 2: mixing up opposite and adjacent. Mark them from the angle you are using. Trap 3: giving the sine as hypotenuse over opposite (upside down).
Trap 4: using degrees or radians wrongly on the calculator. In Bluebook's Desmos calculator, check the mode is set to the right unit. Trap 5: not reading what the question asks: the area, the perimeter or just a side.
Good habit: draw the triangle, write the sides you know, mark the right angle, and label the angle in the question.
See it
Sides are named from the angle θ. SOH: sin = opposite/hypotenuse. CAH: cos = adjacent/hypotenuse. TOA: tan = opposite/adjacent.
| Triangle | Sides |
|---|---|
| 3-4-5 | 3, 4, 5 and multiples (6-8-10, 9-12-15) |
| 5-12-13 | 5, 12, 13 and multiples (10-24-26) |
| 8-15-17 | 8, 15, 17 |
| 7-24-25 | 7, 24, 25 |
| 45-45-90 | x, x, x√2 |
| 30-60-90 | x, x√3, 2x |
- 1Draw it and mark the right angle
- 2Write the sides and angle that are given
- 3Two sides known? Use a² + b² = c² or a triple
- 4An angle is involved? Use SOH-CAH-TOA
- 5Check the hypotenuse is the longest side
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