The most important theorem in geometry — all right triangles obey it
a and b are the legs, c is the hypotenuse (opposite the right angle). Common triples: 3-4-5, 5-12-13, 8-15-17. Multiply by any integer for more triples.
Three basic trig ratios defined from right triangles
SOH: Sine = Opposite ÷ Hypotenuse. CAH: Cosine = Adjacent ÷ Hypotenuse. TOA: Tangent = Opposite ÷ Adjacent. Used to find missing sides and angles.
SOH
Sine = Opposite / Hypotenuse
CAH
Cosine = Adjacent / Hypotenuse
TOA
Tangent = Opposite / Adjacent
Triangle Similarity
Similar triangles: AA (Angle-Angle), SAS~ (Side-Angle-Side similarity), SSS~ (Side-Side-Side similarity) — same shape, different size
Triangle Similarity
Three ways to prove triangles are similar
AA: two pairs of equal angles → similar. SAS~: two proportional sides with equal included angle. SSS~: all three sides proportional. Similar triangles have equal angles and proportional (not equal) sides.
Special Right Triangles
30-60-90: sides are x, x√3, 2x. 45-45-90: legs are x, hypotenuse is x√2.
Special Right Triangles
Two triangle ratios that appear everywhere in geometry and trig
30-60-90: short leg = x, long leg = x√3, hypotenuse = 2x. 45-45-90 (isosceles right): legs = x, hypotenuse = x√2. These ratios recur in trigonometry, calculus, and physics.
30°
Opposite = x
60°
Opposite = x√3
90°
Hypotenuse = 2x
Triangle Inequality Theorem
Triangle inequality: any side must be less than the sum of the other two
Triangle Inequality Theorem
A constraint on what side lengths can form a triangle
For sides a, b, c: a + b > c, a + c > b, b + c > a. If any side ≥ sum of other two, no triangle can be formed. Test: can 3, 4, 8 form a triangle? 3 + 4 = 7 < 8 → NO.
Centroid: intersection of medians (each connects vertex to midpoint of opposite side). Divides each median 2:1 from vertex. Center of gravity. Circumcenter: intersection of perpendicular bisectors — equidistant from all vertices. Center of circumscribed circle. Incenter: intersection of angle bisectors — equidistant from all sides. Center of inscribed circle.
Centroid
Medians meet — center of gravity
Circumcenter
Perpendicular bisectors meet — circumscribed circle center
Incenter
Angle bisectors meet — inscribed circle center
Orthocenter
Altitudes meet
Triangle Line Segments
Median: vertex to midpoint of opposite side. Altitude: perpendicular from vertex to opposite side.
Triangle Line Segments
Four important line segments in a triangle
Median: connects vertex to midpoint of opposite side — three medians always meet at centroid. Altitude: perpendicular segment from vertex to line containing opposite side — can be outside triangle (obtuse). Angle bisector: bisects the angle. Perpendicular bisector: bisects side at 90° — doesn't go through opposite vertex.
Heron's Formula
Heron's formula: Area = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2 is the semi-perimeter
Heron's Formula
Find triangle area when you know all three sides but no height
When you know all three sides (SSS) but not the height, Heron's formula works. s = semi-perimeter = half the perimeter. Area = √[s(s-a)(s-b)(s-c)]. Example: sides 3, 4, 5 → s=6, Area = √[6(3)(2)(1)] = √36 = 6. Confirms ½×3×4=6.
Exterior Angle Theorem
Exterior angle of a triangle = sum of two non-adjacent interior angles
Exterior Angle Theorem
A shortcut that avoids finding the third interior angle
The exterior angle (formed by extending one side) equals the sum of the two non-adjacent (remote) interior angles. If a triangle has angles 40° and 65°, the exterior angle at the third vertex = 40°+65° = 105°. Faster than: find third interior angle (75°), then subtract from 180°.
Triangle Midsegment
Midsegment theorem: segment connecting midpoints of two sides is parallel to third side and half its length
Triangle Midsegment
A segment that connects midpoints creates a miniature similar triangle
Midsegment: connects midpoints of two sides. It is: (1) parallel to the third side, (2) exactly half the length of the third side. The midsegment creates a smaller triangle similar to the original with scale factor ½. Three midsegments divide any triangle into four congruent triangles.
Trigonometric Area Formula
Area with trig: Area = ½ab sinC where a and b are two sides and C is the included angle
Trigonometric Area Formula
Find triangle area using two sides and the included angle
When you know two sides and the angle between them (SAS), use Area = ½ab sinC. Example: sides 8 and 6, included angle 30°. Area = ½(8)(6)sin30° = ½(8)(6)(0.5) = 12. Also useful: derives the Law of Sines from this formula.
Triangle Inequality
Triangle inequality theorem: the sum of any two sides must be GREATER than the third side
Triangle Inequality
A necessary condition for three lengths to form a triangle
For sides a, b, c: a+b>c AND a+c>b AND b+c>a. If any condition fails, no triangle can be formed. Test: 3, 4, 8 → 3+4=7 < 8 → NOT a valid triangle. 5, 7, 9 → 5+7=12>9, 5+9=14>7, 7+9=16>5 → VALID. Also: the largest angle is opposite the longest side.
Mnemonic
What it means
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🎓 Common Exam Questions
Q: Explain SOH CAH TOA — what does each letter stand for and how do you use it?
A: SOH: Sin equals Opposite over Hypotenuse. CAH: Cosine equals Adjacent over Hypotenuse. TOA: Tangent equals Opposite over Adjacent. In a right triangle with angle theta: the opposite side is across from theta, the adjacent side is next to theta (not the hypotenuse), and the hypotenuse is always across from the right angle. To find a missing side: identify which trig ratio involves the known and unknown, set up the equation, solve. Example: angle 30 degrees, hypotenuse = 10, find the opposite side. Sin(30) = opposite/10. Opposite = 10 times sin(30) = 10 times 0.5 = 5.
Q: What are the four triangle centers and where does each lie?
A: Centroid (G): intersection of the three medians (each median connects a vertex to the midpoint of the opposite side). Always INSIDE the triangle. The centroid divides each median 2:1 from vertex to midpoint. It is the center of mass — where the triangle balances. Circumcenter (C): intersection of the three perpendicular bisectors. Center of the circumscribed circle (circumcircle). Can be inside (acute), on the hypotenuse (right), or outside (obtuse) the triangle. Incenter (I): intersection of the three angle bisectors. Always INSIDE. Center of the inscribed circle (incircle). Equidistant from all three sides. Orthocenter (H): intersection of the three altitudes. Inside for acute, on right angle for right, outside for obtuse. Mnemonic: 'Can I Get Help' — Circumcenter, Incenter, Centroid, ortHocenter.
Q: Compare the three triangle similarity shortcuts — AA, SAS~, SSS~.
A: AA (Angle-Angle): if two angles of one triangle equal two angles of another, the triangles are similar. The third pair of angles is automatically equal since angle sums are both 180 degrees. This is the most commonly used similarity shortcut — you only need TWO angle pairs. SAS~ (Side-Angle-Side similarity): two pairs of corresponding sides are proportional AND the included angles are equal. The included angle must be BETWEEN the proportional sides. SSS~ (Side-Side-Side similarity): all three pairs of corresponding sides are proportional. Unlike the congruence shortcuts, AA works as a similarity shortcut (but AAA does not work for congruence). Note: do not confuse with congruence shortcuts — SAS, SSS, ASA, AAS, HL prove congruence (equal size); AA, SAS~, SSS~ prove only similarity (same shape, possibly different size).
Q: State the triangle inequality theorem and explain how to test if three lengths can form a triangle.
A: Triangle inequality theorem: the sum of any two sides of a triangle must be GREATER THAN the third side. This must hold for all three combinations. Equivalently: the longest side must be less than the sum of the other two sides. Test: given sides a, b, c where c is largest — check only if a + b > c (the other two inequalities are automatically satisfied if this one holds). Example: can 4, 7, 12 form a triangle? 4 + 7 = 11 < 12 — No! Can 5, 8, 10 form a triangle? 5 + 8 = 13 > 10 — Yes! Range for a missing side: if two sides are a and b, the third side x must satisfy |a-b| < x < a+b.
Q: Explain the special right triangles — 30-60-90 and 45-45-90 — and how to memorize them.
A: 45-45-90 triangle: an isosceles right triangle. Sides are x, x, x times square root of 2. The hypotenuse is always the leg times square root of 2. Memory: two legs equal (isosceles), hypotenuse gets the square root of 2. 30-60-90 triangle: half of an equilateral triangle. Sides are x (short leg, opposite 30), x times square root of 3 (long leg, opposite 60), 2x (hypotenuse, opposite 90). Memory: short leg doubles to get hypotenuse, multiply short leg by square root of 3 to get long leg. Common Pythagorean triples (right triangle sides): 3-4-5, 5-12-13, 8-15-17, 7-24-25. Multiples also work: 6-8-10 (2 times 3-4-5), 9-12-15 (3 times 3-4-5).