Right triangles support two powerful tools: the Pythagorean theorem for any right triangle and exact side ratios for triangles with 45-45-90 or 30-60-90 angles.
Right-triangle anatomy
Right-triangle anatomyA right triangle has perpendicular legs a and b, hypotenuse c opposite the right angle, and two complementary acute angles.
Hypotenuse
The side opposite the (90^\circ) angle. It is the longest side and must be (c) in (a^2+b^2=c^2).
Pythagorean theorem and converse
Pythagorean theorem
\[a^2+b^2=c^2\]
Use only for a right triangle, with (c) opposite the right angle.
A 9-12-15 right triangleA right triangle has legs 9 and 12 and hypotenuse 15, demonstrating 9 squared plus 12 squared equals 15 squared.
Forward theorem versus converse
Known right triangle
Use (a^2+b^2=c^2) to calculate a missing side.
Known three side lengths
Put the longest length in the (c) position. Equality proves the triangle is right; inequality means it is not right.
Worked example
Missing leg
A right triangle has hypotenuse 17 and one leg 8. Find the other leg.
Identify c
The hypotenuse is (c=17).
Subtract
(a^2=17^2-8^2=289-64=225).
Root
A length is positive, so (a=\sqrt{225}=15).
The missing leg is (15).
Special right triangles
45-45-90 triangleAn isosceles right triangle has two 45-degree angles, legs x and x, and hypotenuse x square root of 2.30-60-90 triangleA right triangle has a 30-degree angle opposite short leg x, a 60-degree angle opposite long leg x square root of 3, and hypotenuse 2x.
Special-right-triangle comparison
Triangle
Angles
Side ratio
Key relationship
45-45-90
(45^\circ,45^\circ,90^\circ)
(1:1:\sqrt2)
hypotenuse = leg(\sqrt2)
30-60-90
(30^\circ,60^\circ,90^\circ)
(1:\sqrt3:2)
hypotenuse = (2)(short leg)
Why 45-45-90 contains square root of 2
\[x^2+x^2=c^2\Rightarrow c=x\sqrt2\]
Two connected right trianglesAltitude BD divides triangle ABC into two right triangles. Solve one triangle for BD, then use BD in the second triangle.
Figure not drawn to scale.
Composite workflow
First triangle
Use the given sides to solve the first right triangle.
Shared result
Carry the shared altitude or diagonal into the second triangle.
Second triangle
Apply Pythagorean or a special ratio again.
Mini check
Choose the correct ratio
A 30-60-90 triangle has short leg 7. What is the longer leg?
Show answer and explanation
Answer: (7\sqrt3).
The longer leg is short leg times (\sqrt3), not times 2.
Key takeaways
Key takeaways
The hypotenuse is opposite the right angle.
Square lengths before adding or subtracting, then take the positive square root.
45-45-90 uses (1:1:\sqrt2).
30-60-90 uses (1:\sqrt3:2).
Continue learning
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.