Demonstration 1 of 4
Legs, hypotenuse and the quick check
Once you know the two legs, how long is the side across from the right angle?
The legs are squared and added; the hypotenuse is the length whose square matches that sum. The quick check says it must beat the longer leg but lose to the walk around the corner.
\[a^2+b^2=c^2\]
\[\max(a, b) < c < a + b\]
a and b are the legs, the two sides that meet at the right angle. c is the hypotenuse, the side opposite it. max(a, b) means the larger leg. All lengths are in grid units.
Predict first. Double both legs from 3 and 4 to 6 and 8. Does the hypotenuse double too?
Choose an example
Constructed example: the chapter's 3, 4, 5 grid triangle, the 6, 8, 10 grid and the 9, 12, 15 ramp.
Calculated values
- Legs
- 3 and 4
- Sum of squared legs
- 3² + 4² = 9 + 16 = 25
- Hypotenuse
- sqrt(25) = 5
- Two-leg route
- 3 + 4 = 7
- Magnitude check
- 4 < 5 < 7
The legs meet at the marked right angle. 3 x 3 + 4 x 4 = 9 + 16 = 25, and 5 x 5 = 25, so the hypotenuse is 5. It is longer than the longer leg, 4, and shorter than the route around the corner, 3 + 4 = 7.
Use the idea
Before trusting any diagonal, check it sits between the longer leg and the sum of the legs. An answer outside that window is wrong before any decimals.
Where the conclusion applies
A flat grid and a stated right angle where the legs meet. Without that right angle, the relationship does not apply.
Check your understanding: Legs of 5 and 12 give what hypotenuse, and does it pass the quick check?
Chapter 18 source: section "Find the right angle first". Demonstration C18-D01.
Demonstration 2 of 4
Four triangles explain the squares
Why does the rule add the squares of the legs instead of the legs themselves?
The big square is counted two ways: by its side, and by its pieces. Take away the four triangles and the middle square is all that remains, which is a² + b².
\[(a+b)^2=4\left(\frac{ab}{2}\right)+c^2=2ab+c^2.\]
a and b are the legs of each corner triangle, c is its hypotenuse. (a + b)² is the big square's area, ab/2 is one triangle's area, and c² is the tilted middle square.
Predict first. Shrink the short leg a from 3 to 1 while b stays 4. What area is left in the middle?
Choose an example
Constructed example: the chapter's four-triangle square, with the book's legs 3 and 4 and other leg lengths.
Calculated values
- Big square (a + b)²
- (3 + 4)² = 49
- Four triangles 2ab
- 4 x 3 x 4 / 2 = 24
- Central square c²
- 49 - 24 = 25
- a² + b²
- 9 + 16 = 25
Count the big square two ways. Its side is 3 + 4 = 7, so its area is 7 x 7 = 49. The four triangles cover 4 x 3 x 4 / 2 = 24. What is left is the central square: 49 - 24 = 25, which matches 3 x 3 + 4 x 4 = 9 + 16 = 25.
Use the idea
When a formula seems handed down from nowhere, count one area in two ways. If both counts agree, the formula has earned its place.
Where the conclusion applies
Four identical right triangles in a flat plane, placed so each outer side is a + b. If the triangles were not right, the middle shape would not be a square.
Check your understanding: With legs 5 and 5, what are the big square, the four triangles and the middle square?
Chapter 18 source: section "Four triangles explain the squares". Demonstration C18-D02.
Demonstration 3 of 4
Trap a square root that is not whole
When the squares add to 20, how do you find the length without a calculator?
Square nearby numbers until one lands below the target and the next lands above. The root is trapped between them; finer steps shrink the trap.
\[c^2=2^2+4^2=20\]
\[4.47^2=19.9809\]
c is the hypotenuse in meters. sqrt(20) is the exact length whose square is 20. A bracket is a pair of numbers, one squaring below the target and one above.
Predict first. Switch from whole numbers to tenths. Which two tenths trap sqrt(20)?
Choose an example
Constructed example: the chapter's 2 and 4 meter legs, with other leg pairs.
Calculated values
- c² for these legs
- 2² + 4² = 4 + 16 = 20
- Exact hypotenuse
- sqrt(20) m
- Bracket
- 4.4 < sqrt(20) < 4.5
- Nearest hundredth
- about 4.47 m
2 x 2 + 4 x 4 = 4 + 16 = 20, so c = sqrt(20) m exactly. Squaring nearby numbers: 4.4 x 4.4 = 19.36 is below 20 and 4.5 x 4.5 = 20.25 is above it, so the root lies between 4.4 and 4.5. To the nearest hundredth it is about 4.47 m; that decimal is a rounded display, not the exact value.
Use the idea
Keep the exact root for later work, and give a rounded decimal only when someone needs a display.
Where the conclusion applies
A stated right angle and legs in meters. The decimal is rounded, so its square need not equal the target exactly. When the root is whole, the trap closes on it.
Check your understanding: Legs of 1 and 3 give c² = 10. Between which two tenths does sqrt(10) lie?
Chapter 18 source: section "Worked example: when the square root is not whole". Demonstration C18-D03.
Demonstration 4 of 4
When the right angle is missing
A triangle has sides 4, 5 and 6. Is it a right triangle?
Every right triangle must make the two bars equal. If they differ, the triangle is not right, and the given third side is kept, never swapped for a new one.
\[4² + 5² = 16 + 25 = 41\]
\[6² = 36\]
The three numbers are given side lengths, with the longest last. Each square is that side times itself.
Predict first. Pick sides 3, 4 and 5. Do the two bars match this time?
Choose an example
Constructed example: the chapter's 4, 5, 6 triangle, its 3, 4, 5 triangle, and other stated sides.
Calculated values
- Shorter sides squared
- 4² + 5² = 16 + 25 = 41
- Longest side squared
- 6² = 36
- Comparison
- unequal (41 > 36)
- Right triangle?
- no, ruled out
With sides 4, 5 and 6, 4 x 4 + 5 x 5 = 16 + 25 = 41 and 6 x 6 = 36. 41 is not 36, so this triangle is not right. The third side stays 6; it is never replaced by sqrt(41).
Use the idea
Before reaching for the rule, ask where the right angle came from: a marking, a statement or the axes. A drawing that only looks square is not enough.
Where the conclusion applies
Three stated side lengths in a flat plane. Unequal bars rule out a right angle. Equal bars are not used here to prove one; that condition must be supplied.
Check your understanding: Is a triangle with sides 6, 7 and 9 a right triangle?
Chapter 18 source: section "When the condition is missing". Demonstration C18-D04.