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#deck:Math::Pythagorean theorem
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What is a right triangle?	A triangle with one angle of 90 degrees, a square corner.	vocabulary
Can a triangle have two right angles?	No. The three angles add up to 180 degrees, so two right angles would leave 0 degrees for the third.	vocabulary
What is the hypotenuse?	The side opposite the right angle in a right triangle. It is always the longest side.	vocabulary
What are the legs of a right triangle?	The two sides that form the right angle.	vocabulary
What does the small square drawn in a corner of a triangle mean?	That the angle there is a right angle, 90 degrees.	vocabulary
State the Pythagorean theorem.	In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs: c^2 = a^2 + b^2.	theorem
In c^2 = a^2 + b^2, which letter must be the hypotenuse?	c, the side opposite the right angle. a and b are the legs, in either order.	theorem
State the converse of the Pythagorean theorem.	If the square of one side of a triangle equals the sum of the squares of the other two, the triangle is a right triangle, and the right angle is opposite that side.	converse
What is the converse of a statement if A then B?	If B then A. It is a different claim and must be proved separately.	converse
Give an example of a true statement whose converse is false.	If it is a dog, it has four legs. The converse, if it has four legs it is a dog, is false.	converse
What does the theorem let you do, and what does the converse let you do?	The theorem finds a missing side when you know the angle is right. The converse proves the angle is right when you know the three sides.	converse
Testing three lengths for a right triangle: what is the first step?	Find the longest side. It is the only candidate for the hypotenuse.	converse
Why check only the longest side as the hypotenuse?	A right angle would be opposite the hypotenuse, and the hypotenuse is always the longest side. With a leg in c's place, even a true right triangle fails.	converse
Sides 6, 4, 3: right triangle?	No. The longest is 6, and 36 is not 16 + 9 = 25.	problems
Sides 5, 12, 13: right triangle?	Yes. 13^2 = 169 = 25 + 144. The hypotenuse is 13.	problems
Sides 7, 24, 25: right triangle?	Yes. 25^2 = 625 = 49 + 576. The hypotenuse is 25.	problems
Sides 4, 5, 6: right triangle?	No. 36 is less than 16 + 25 = 41, so the largest angle is acute.	problems
Sides 2, 2, 3: right triangle?	No. 9 is more than 4 + 4 = 8, so the largest angle is obtuse.	problems
Legs 4 and 3: hypotenuse?	5, since 16 + 9 = 25.	problems
Legs 5 and 12: hypotenuse?	13, since 25 + 144 = 169.	problems
Legs 14 and 48: hypotenuse?	50, since 196 + 2304 = 2500. It is 7-24-25 doubled.	problems
Hypotenuse 13 and one leg 5: the other leg?	12, since 169 - 25 = 144. Subtract from the hypotenuse's square.	problems
Legs 1 and 1: hypotenuse?	The square root of 2, about 1.4142. Exact only as a square root.	problems
If c^2 is less than a^2 + b^2, with c the longest side, what is the angle opposite c?	Acute, less than 90 degrees.	acute-obtuse
If c^2 is greater than a^2 + b^2, with c the longest side, what is the angle opposite c?	Obtuse, more than 90 degrees.	acute-obtuse
What is the triangle inequality?	Each side of a triangle is shorter than the other two added together. If not, the three lengths make no triangle.	acute-obtuse
Sides 3, 4, 7: what kind of triangle?	None. 3 + 4 = 7, so the three lengths lie flat along a line.	acute-obtuse
Which general law has the Pythagorean theorem as its 90 degree case?	The law of cosines, c^2 = a^2 + b^2 - 2ab cos C. At C = 90 degrees cos C = 0.	acute-obtuse
Rearrangement proof: what does a square of side a + b contain?	Four copies of the right triangle, area ab/2 each, and a tilted square of side c in the middle.	proof
Finish the rearrangement proof from (a + b)^2 = 4(ab/2) + c^2.	Expand: a^2 + 2ab + b^2 = 2ab + c^2. Subtract 2ab: a^2 + b^2 = c^2.	proof
What is a Pythagorean triple?	Three whole numbers a, b, c with a^2 + b^2 = c^2, such as 3, 4, 5.	triples
Name five Pythagorean triples worth knowing by sight.	3-4-5, 5-12-13, 8-15-17, 7-24-25, 20-21-29.	triples
If 3, 4, 5 is a triple, why is 6, 8, 10?	Multiply every side by k and every square by k^2, so the equation still holds.	triples
From c^2 = 25, why is c = 5 and not -5?	c is a length, and lengths are not negative. The square root sign means the non-negative root.	theorem
Burj Khalifa example: which angle is the right angle?	Where the line of sight touches the Earth: a tangent line meets the radius at 90 degrees.	problems
Burj Khalifa example: why is (R + h)^2 - R^2 risky on a calculator?	It subtracts two nearly equal numbers, about 15.7 million each, which cancels their leading digits. Use 2Rh + h^2 instead.	precision
How far can you see from 1483 feet up, with Earth's radius 3960 miles?	About 47 miles: d is the square root of 2Rh + h^2, about 2224.58 square miles.	problems
Trap: legs 3 and 4, so c = 3 + 4 = 7?	No. Squares add, not sides: c^2 = 9 + 16 = 25, c = 5. 3 + 4 = 7 would be a flat line.	trap
Trap: hypotenuse 10, leg 6, so the other leg is the square root of 100 + 36?	No. Subtract when c is known: 100 - 36 = 64, leg 8. Adding would make a leg longer than the hypotenuse.	trap
Trap: sides 8, 15, 17 listed as 17, 8, 15. Test 15^2 against 17^2 + 8^2?	No. Sort first: 17 is longest. 289 = 64 + 225, so it is right, with hypotenuse 17.	trap
Trap: c^2 = a^2 + b^2 holds for every triangle if you pick the right labels?	No. It holds only for right triangles. For 4, 5, 6 no labelling works.	trap
Trap: is the square root of (a^2 + b^2) equal to a + b?	No. The square root of 9 + 16 is 5, not 7. A square root does not split over a sum.	trap
Trap: a triangle has sides 1, 1 and the square root of 2. Which angle is right?	The one opposite the square root of 2, the longest side: 1 + 1 = 2.	trap
Trap: every triangle with c^2 greater than a^2 + b^2 is obtuse. True for any labels?	Only when c is the longest side. With a short side as c, the inequality says nothing about the largest angle.	trap
Trap: the theorem proves a triangle with sides 5, 12, 13 is right. Theorem or converse?	The converse. The theorem assumes the right angle; the converse concludes it from the sides.	trap
What does perpendicular mean?	Two lines meeting at a right angle; all four angles where they cross are 90 degrees. Written with the symbol that looks like an upside-down T.	vocabulary
A right angle in degrees, radians and gradians?	90 degrees, pi/2 radians, 100 gradians: a quarter turn in each.	vocabulary
The two acute angles of a right triangle add up to what?	90 degrees: they are complementary. If one is theta, the other is 90 - theta.	theorem
Why is the hypotenuse the longest side?	The largest angle of a triangle is opposite the longest side, and the right angle is the largest.	theorem
Why is the hypotenuse shorter than the sum of the legs?	(a + b)^2 = a^2 + b^2 + 2ab, which is more than c^2 by 2ab.	theorem
Side ratios of a 45-45-90 triangle, and where they come from?	1 : 1 : square root of 2. Half a square cut along its diagonal: c^2 = 1 + 1.	special-triangles
Side ratios of a 30-60-90 triangle, and where they come from?	1 : square root of 3 : 2. Half an equilateral triangle of side 2: long leg^2 = 4 - 1 = 3.	special-triangles
In a 30-60-90 triangle, which side is half the hypotenuse?	The short leg, opposite the 30 degree angle.	special-triangles
45-45-90 with legs 7: hypotenuse?	7 times the square root of 2, about 9.90.	special-triangles
30-60-90 with hypotenuse 10: legs?	5 and 5 times the square root of 3, about 8.66.	special-triangles
Space diagonal of a box p by q by r?	The square root of p^2 + q^2 + r^2: Pythagoras on a face, then again with the third edge.	three-d
Space diagonal of a 2 by 3 by 6 box?	7, since 4 + 9 + 36 = 49.	three-d
Diagonal of a unit cube?	The square root of 3.	three-d
State the law of cosines.	c^2 = a^2 + b^2 - 2ab cos C, for any triangle, with C the angle opposite c.	law-of-cosines
How does the law of cosines reduce to Pythagoras?	At C = 90 degrees cos C = 0, so the last term vanishes.	law-of-cosines
Trap: Pythagoras is a consequence of the law of cosines?	No, the other way round. The law of cosines is usually proved with Pythagoras; Pythagoras is the special case, not the consequence.	trap
Trap: in a 30-60-90 triangle, the long leg is half the hypotenuse?	No. The short leg, opposite 30 degrees, is half. The long leg is the square root of 3 times the short leg.	trap
Trap: the hypotenuse is the denominator of every basic trig ratio?	No. Tangent is opposite over adjacent. Sine and cosine divide by the hypotenuse.	trap
Why is the converse of the Pythagorean theorem true?	Build a right triangle with legs a and b. Its hypotenuse is the square root of a^2 + b^2 = c, so both triangles have sides a, b, c, and SSS makes them congruent: the right angle carries over.	converse
Which congruence rule proves the converse of the Pythagorean theorem?	Side-Side-Side.	converse
Trap: the right angle is opposite whichever side is alone on one side of the equation?	Only if you wrote it as a^2 + b^2 = c^2. In c^2 - b^2 = a^2 a leg is alone. Safe rule: the right angle is opposite the longest side.	trap
In a^2 + b^2 = c^2, is c a name or a job?	A job: the side opposite the right angle. The equation says (leg)^2 + (other leg)^2 = (hypotenuse)^2.	converse
In the Pythagorean equation, which side stands alone and which stand together?	The longest side, squared, stands alone. The other two, squared and added, stand together on the other side.	converse
Does it matter whether c^2 is on the left or the right of the equals sign?	No. a^2 + b^2 = c^2 and c^2 = a^2 + b^2 are the same equation read both ways. What matters is that c^2 is alone.	converse
The three steps of the Pythagorean test?	1. Find the longest side, c. 2. Put c^2 alone on one side, the other two squares added on the other. 3. Compare.	converse
Why must the side standing alone be the longest?	Two squares added are bigger than either one alone, so the lone square must be the biggest.	converse
3-4-5 with 4 standing alone: 3^2 + 5^2 against 4^2?	34 against 16: not equal. Same right triangle, wrong setup: it claims leg^2 + hypotenuse^2 = other leg^2.	trap
3-4-5 with 3 standing alone: 4^2 + 5^2 against 3^2?	41 against 9: not equal. A short side alone always leaves the lone square too small.	trap
Trap: 3^2 + 5^2 = 34 is not 4^2 = 16, so the 3-4-5 triangle is not right?	No. The triangle is right; the setup is wrong. The longest side, 5, must stand alone: 9 + 16 = 25.	trap
Trap: c^2 = a^2 + b^2 is wrong because c must be on the right?	No. An equation reads both ways. Only alone and together matter, not left and right.	trap