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Pythagorean Theorem
a² + b² = c² — legs squared = hypotenuse squared
Pythagorean Theorem
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.
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🃏 Pythagorean Theorem
The Pythagorean theorem?
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🃏 Answer
a² + b² = c² — legs squared = hypotenuse squared
Pythagorean Theorem — 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.
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Trig Ratios
SOH CAH TOA (SOH=Sine Opposite Hypotenuse, CAH=Cosine Adjacent Hypotenuse, TOA=Tangent Opposite Adjacent): Sin=Opp/Hyp, Cos=Adj/Hyp, Tan=Opp/Adj
Trig Ratios
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
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🃏 Trig Ratios
SOH CAH TOA
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🃏 Answer
SOH CAH TOA (SOH=Sine Opposite Hypotenuse, CAH=Cosine Adjacent Hypotenuse, TOA=Tangent Opposite Adjacent): Sin=Opp/Hyp, Cos=Adj/Hyp, Tan=Opp/Adj
SOHSine = Opposite / Hypotenuse
CAHCosine = Adjacent / Hypotenuse
TOATangent = Opposite / Adjacent
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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.
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🃏 Triangle Similarity
Triangle similarity — the three shortcuts?
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🃏 Answer
Similar triangles: AA (Angle-Angle), SAS~ (Side-Angle-Side similarity), SSS~ (Side-Side-Side similarity) — same shape, different size
Triangle Similarity — 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.
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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
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🃏 Special Right Triangles
Special right triangles — the 30-60-90 and 45-45-90 side ratios?
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🃏 Answer
30-60-90: sides are x, x√3, 2x. 45-45-90: legs are x, hypotenuse is x√2.
30°Opposite = x
60°Opposite = x√3
90°Hypotenuse = 2x
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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.
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🃏 Triangle Inequality Theorem
The triangle inequality theorem?
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🃏 Answer
Triangle inequality: any side must be less than the sum of the other two
Triangle Inequality Theorem — 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.
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Triangle Centers
Triangle centers: centroid (medians), circumcenter (perpendicular bisectors), incenter (angle bisectors), orthocenter (altitudes)
Triangle Centers
Four special points every triangle has
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
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🃏 Triangle Centers
The four triangle centers — which lines make each?
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🃏 Answer
Triangle centers: centroid (medians), circumcenter (perpendicular bisectors), incenter (angle bisectors), orthocenter (altitudes)
CentroidMedians meet — center of gravity
CircumcenterPerpendicular bisectors meet — circumscribed circle center
IncenterAngle bisectors meet — inscribed circle center
OrthocenterAltitudes meet
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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.
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🃏 Triangle Line Segments
Median vs altitude?
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Median: vertex to midpoint of opposite side. Altitude: perpendicular from vertex to opposite side.
Triangle Line Segments — 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.
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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.
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🃏 Heron's Formula
Heron's formula?
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🃏 Answer
Heron's formula: Area = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2 is the semi-perimeter
Heron's Formula — 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.
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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°.
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🃏 Exterior Angle Theorem
The exterior angle theorem?
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🃏 Answer
Exterior angle of a triangle = sum of two non-adjacent interior angles
Exterior Angle Theorem — 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°.
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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.
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🃏 Triangle Midsegment
The midsegment theorem?
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🃏 Answer
Midsegment theorem: segment connecting midpoints of two sides is parallel to third side and half its length
Triangle Midsegment — 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.
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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.
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🃏 Trigonometric Area Formula
Triangle area using trigonometry?
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🃏 Answer
Area with trig: Area = ½ab sinC where a and b are two sides and C is the included angle
Trigonometric Area Formula — 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.
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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.
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🃏 Triangle Inequality
Triangle inequality — what must be true of the sides?
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🃏 Answer
Triangle inequality theorem: the sum of any two sides must be GREATER than the third side
Triangle Inequality — 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.
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