What is the Pythagorean theorem?+
For a right triangle with legs a, b and hypotenuse c: a² + b² = c². It applies ONLY to right (90°) triangles. Any triangle satisfying this equation is guaranteed to be a right triangle (converse theorem). Example: 3² + 4² = 9 + 16 = 25 = 5², confirming 3-4-5 is a right triangle.
How do I find a missing angle in a right triangle?+
Use inverse trig: A = arcsin(opposite/hypotenuse), arccos(adjacent/hypotenuse), or arctan(opposite/adjacent). Since one angle is already 90°, the third angle = 90° − A. Example: if a=3, c=5, then A = arcsin(3/5) = arcsin(0.6) ≈ 36.87°, so B ≈ 53.13°.
What are the 30-60-90 triangle ratios?+
Sides are in ratio 1 : √3 : 2. If the short leg (opposite 30°) = n, the long leg (opposite 60°) = n√3, and hypotenuse = 2n. Derived by bisecting an equilateral triangle. Used in architecture, roof framing, and tiling. Example: short leg = 5 → long leg ≈ 8.66, hypotenuse = 10.
What are the 45-45-90 triangle ratios?+
Sides are in ratio 1 : 1 : √2. Both legs equal; the hypotenuse = leg × √2. Appears when cutting a square diagonally. If the leg = 5 cm, hypotenuse = 5√2 ≈ 7.07 cm. Both acute angles are exactly 45°.
What is the area of a right triangle?+
Area = (1/2) × a × b, where a and b are the two legs (the sides forming the right angle). This is a special case of A = (1/2) × base × height, because in a right triangle, the two legs serve directly as base and height. Example: legs 6 and 8 → Area = (1/2)(6)(8) = 24 sq units.
Can I solve a triangle knowing only three sides (SSS)?+
Yes. With all three sides, compute angles using inverse cosine: A = arccos((b²+c²−a²)/(2bc)). For a right triangle, the hypotenuse is opposite the 90° angle. Verify first: if the longest side² equals the sum of squares of the other two, you have a right triangle.
What is the law of sines?+
For any triangle: a/sin(A) = b/sin(B) = c/sin(C). Useful for AAS and ASA cases. For right triangles, since sin(90°) = 1, the law simplifies to a/sin(A) = c, confirming sin(A) = a/c (SOHCAHTOA). The law of sines cannot uniquely solve SSA configurations (the ambiguous case).
What does SOHCAHTOA mean?+
SOHCAHTOA is a memory aid for right-triangle trig ratios: SOH = Sine × Opposite/Hypotenuse; CAH = Cosine × Adjacent/Hypotenuse; TOA = Tangent × Opposite/Adjacent. Example: angle A, opposite=4, adjacent=3, hypotenuse=5 → sin A = 4/5 = 0.8, cos A = 3/5 = 0.6, tan A = 4/3 ≈ 1.333.
What is a Pythagorean triple?+
Three positive integers (a, b, c) satisfying a² + b² = c². Common triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 20-21-29. All multiples also work: 6-8-10, 9-12-15. In construction, a 3-4-5 string triangle is used to verify right angles without a protractor (measure 3 units, 4 units, and confirm the diagonal is 5 units).
How is the circumradius of a right triangle calculated?+
R = c/2, where c is the hypotenuse. By Thales' theorem, the hypotenuse is always the diameter of the circumscribed circle. If hypotenuse = 10 cm, then R = 5 cm. The right-angle vertex always lies on the circumscribed circle. This property is used in circle geometry proofs and engineering design.
What is the inradius of a right triangle?+
r = (a + b − c) / 2, where a and b are legs and c is the hypotenuse. The incircle touches all three sides. Example: 3-4-5 triangle → r = (3 + 4 − 5)/2 = 1 unit. The inradius is used in structural design and pipe-fitting problems (fitting a circular pipe inside a triangular space).
What is the altitude to the hypotenuse?+
h = (a × b) / c, where a and b are the legs and c the hypotenuse. The altitude from the right-angle vertex to the hypotenuse divides the triangle into two smaller triangles, each similar to the original. Geometric mean relationship: h² = p × q, where p and q are the two hypotenuse segments. Used in lens design and optical path calculations.