📐 Geometry · Similarity and Right Triangles
The Pythagorean Theorem: fun ways and memory help
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
Cutting across the field
A coach marks a rectangular practice field that is 30 yards long and 20 yards wide. Two players start at the same corner and race to the opposite corner. One runs along the two sides, which is 30 + 20 = 50 yards. The other runs straight across the field, which is √(302 + 202) = √1300 = 10√13 ≈ 36.06 yards.
How it connects: The two sides are the legs and the straight run is the hypotenuse, so the straight run is c = √(a2 + b2), not a + b. It is shorter because c2 = a2 + b2 is less than (a + b)2. The limit: the corner of a rectangle is a right angle. If the field is not a rectangle, or something blocks the way, this formula does not apply.
Square pizzas
A pizza shop sells square pizzas cut into square slices. A 3-by-3 pizza has 9 slices, a 4-by-4 pizza has 16 slices, and a 5-by-5 pizza has 25 slices. Two friends order a 3-by-3 and a 4-by-4 pizza. Together they get 9 + 16 = 25 slices, exactly as many as one 5-by-5 pizza.
How it connects: The slice counts are the areas of squares with sides 3, 4 and 5, which are the squares on the sides of a right triangle: 32 + 42 = 52. The limit: this works only because 3, 4, 5 are the sides of a right triangle. Sides do not add: a 3-by-3 and a 4-by-4 pizza do not make a 7-by-7 pizza (49 slices). The areas add, and the square with area 25 has side 5. One example does not prove the rule for every triangle; the two-arrangement picture does that.
Measure twice, cut once
This saying means: check your measurements carefully before doing something you cannot undo, such as cutting a board.
How it connects: A builder can check that a corner is square with the converse. Mark 3 feet along one edge and 4 feet along the other. If the straight-line distance between the two marks is exactly 5 feet, then 32 + 42 = 52, so the corner is a right angle. Measuring that third side is the second measurement. The limit: the check is only as good as the measuring. A larger multiple such as 6, 8, 10 also works, and a small slip in measuring matters less.
As the crow flies
This idiom means the straight-line distance between two places, ignoring roads and obstacles. Suppose Town B is 7 miles east and 10 miles north of Town A, and the roads run only east-west and north-south, so driving is 7 + 10 = 17 miles. As the crow flies, the distance is √(72 + 102) = √149 ≈ 12.21 miles.
How it connects: The two road segments are the legs and the crow’s path is the hypotenuse: c = √(a2 + b2). The limit: the formula needs the two road directions to meet at a right angle on flat ground. Over very long distances the curve of the Earth matters, and a flat right triangle is no longer exact.
The tower rope
In a made-up building game, a player ties a rope from the top of a tower that is 6 blocks tall to a stake in flat ground 13 blocks from the base of the tower. The rope is pulled tight, so how long must it be? The rope length is √(62 + 132) = √205 ≈ 14.32 blocks.
How it connects: The tower and the ground meet at a right angle, so the rope is the hypotenuse: 62 + 132 = c2. A reasonableness check: 14.32 is longer than 13 and shorter than 6 + 13 = 19. The limit: a loose rope would sag and need more length. The formula gives the length of a tight, straight rope.
Always right
Why do right triangles win every argument? Because they are always right.
How it connects: The pun uses two meanings of “right”. A right angle is exactly 90°, and the Pythagorean theorem is true for every right triangle. The limit: a triangle without a right angle is not “wrong”; the theorem just does not apply to it. And a right angle does not make anyone’s argument correct.
Memory help
Spot, Square, Combine, Root
For a missing side, say Spot, Square, Combine, Root. Spot the hypotenuse (across from the right angle). Square the two sides you know. Combine: add the squares if the hypotenuse is the unknown, or subtract the smaller square from the larger if a leg is the unknown. Root: take the positive square root. Say it as “Silly Seals Climb Rocks.”
Add for long, subtract for short
Add for the long side, subtract for the short, and the root at the end is the last support. Here “long side” means the hypotenuse and “short side” means a leg, since a leg is always shorter than the hypotenuse.
Look across the corner
Stand in the right-angle corner and look straight across. The side you are facing is the hypotenuse. The two sides you could touch with your hands are the legs.
The triple family
3, 4, 5 is the parent: doubled it becomes 6, 8, 10. For 5, 12, 13 think “five fingers, a dozen eggs, a baker’s dozen (13).” In 3, 4, 5 and 5, 12, 13 the hypotenuse is 1 more than the longer leg, and in 8, 15, 17 it is 2 more. That pattern is just a memory help for these three; it does not hold for every triple.
Not c, but c squared
For legs 3 and 4, 32 + 42 = 25. The 25 is c2, not c. Stop there and you are only halfway: c = √25 = 5. If an answer is longer than the sum of the legs, check that the root was taken.
The longest side goes alone
In the test a2 + b2 = c2, the c is the longest side. Squares of the two shorter sides go together on one side of the equals sign, and the square of the longest side goes by itself on the other.
Draw the three squares
Sketch a right triangle and put a square on each side. Label the areas a2, b2 and c2. The biggest square is always on the side across from the right angle, and the two smaller squares together fill the same area.