Right triangles and the Pythagorean theorem — Digital SAT Math practice
Pythagorean theorem, special right triangles.
Key terms
- Hypotenuse: Longest side of a right triangle, always opposite the 90 degree angle.
- Leg: Either of the two shorter sides that form the right angle in a right triangle.
- Pythagorean Triple: Three positive integers that satisfy a^2 + b^2 = c^2, like 3, 4, and 5.
- Special Right Triangle: A right triangle with predictable side ratios, specifically 45-45-90 or 30-60-90.
Common traps
- Assuming the Pythagorean theorem applies to any triangle instead of only right triangles.
- Forgetting that the hypotenuse must always be the longest side and opposite the right angle.
- Mixing up the leg ratios for 30-60-90 triangles by assigning √3 to the shorter leg instead of the longer one.
Key ideas to know
Pythagorean Theorem Application
A right triangle has legs measuring 6 and 8 units. What is the length of the hypotenuse?
Use a2 + b2 = c2. Calculate 62 + 82 = 36 + 64 = 100. The hypotenuse is the square root of 100, which equals 10.
Watch out: Squaring only one leg or forgetting to take the square root at the end.
30-60-90 Triangle Ratios
In a 30-60-90 triangle, the hypotenuse measures 10 units. What are the lengths of the shorter leg and the longer leg?
The sides follow the ratio 1 : sqrt(3) : 2. Divide the hypotenuse (2 parts) by 2 to get the shorter leg (1 part): 10 / 2 = 5. Multiply the shorter leg by sqrt(3) for the longer leg: 5sqrt(3).
Watch out: Swapping the legs or multiplying the hypotenuse by sqrt(3) directly instead of dividing it first.
Recognizing Pythagorean Triples
A right triangle has a hypotenuse of 26 and one leg of 10. What is the length of the other leg?
Look for multiples of common triples. The base triple 5-12-13 multiplied by 2 gives 10-24-26. Since one leg is 10, the missing leg corresponds to the 12 in the triple, so it is 24.
Watch out: Trying to calculate sqrt(262 - 102) manually when recognizing the scaled triple is faster and less error-prone.
Sample questions
Question 1
A right triangle has legs of length 9 and 12. What is the perimeter of the triangle?
- 21
- 30
- 36
- 54
Show answer and explanation
Answer: 36
The hypotenuse is sqrt(9^2 + 12^2) = 15, so the perimeter is 9 + 12 + 15 = 36.