The Lumenary · Grades 6–8

The Pythagorean theorem

Name: ______________________

Date: ______________________

You will learn to

Explain a rearrangement proof of the Pythagorean theorem, including why the pieces fit together
Use the theorem and square roots to find missing side lengths
Use the converse to test whether a triangle has a right angle
Compute the distance between two points in the coordinate plane
Generate Pythagorean triples with Euclid's formula and explain why there are infinitely many

1.
A right triangle has legs $8$ cm and $15$ cm. How long is the hypotenuse, in cm?
2.
A right triangle has hypotenuse $13$ and one leg $5$ . How long is the other leg?
3.
What is the straight-line distance between the points $(1, 5)$ and $(7, -3)$ ?
4.
Which of these triangles has a right angle?
- A.Sides $7$ , $24$ , $25$
- B.Sides $4$ , $5$ , $6$
- C.Sides $6$ , $7$ , $9$
- D.Sides $2$ , $3$ , $4$
5.
A right triangle has one leg $6$ and hypotenuse $10$ . Rahul writes $6^2 + 10^2 = 136$ and says the third side is $\sqrt{136} \approx 11.7$ . What went wrong?
- A.He treated $10$ as a leg. The hypotenuse is the longest side, so the third side satisfies $6^2 + b^2 = 10^2$ and is $8$ .
- B.Nothing — the third side is $\sqrt{136}$ .
- C.He should have added the lengths: the third side is $16$ .
- D.The theorem only works for whole-number triangles.
6.
In the first rearrangement proof, why is the shape in the middle of the left picture a square?
- A.Its four sides are all $c$ , and each of its angles is $180^\circ - (\alpha + \beta) = 90^\circ$ , because $\alpha + \beta = 90^\circ$ in a right triangle.
- B.Because its four sides are all equal to $c$ .
- C.Because it looks like a square in the picture.
- D.Because the big outer shape is a square.
7.
Use Euclid's formula $(m^2 - n^2,\ 2mn,\ m^2 + n^2)$ with $m = 5$ and $n = 2$ . What is the hypotenuse of the triple you get?
8.
A triangle has sides $7$ , $8$ and $11$ . What kind of angle is opposite the side $11$ ?
- A.Acute (less than $90^\circ$ )
- B.A right angle
- C.Obtuse (more than $90^\circ$ )
9.
In the Lean file, sqDist A B is the square of the distance $AB$ and IsRightAngleAt A B C says the angle at $C$ is a right angle. Which statement is the converse of the Pythagorean theorem?
- A.A ≠ C → sqDist A B = sqDist C A + sqDist C B → IsRightAngleAt A B C
- B.IsRightAngleAt A B C → sqDist A B = sqDist C A + sqDist C B
- C.sqDist A B = sqDist C A + sqDist C B
- D.IsRightAngleAt A B C → sqDist A B = sqDist C A * sqDist C B
10.
Write a function is_right(a, b, c) that returns True if a triangle with sides a, b, c (given in any order) has a right angle, and False otherwise. Use whole numbers, so the test is exact.
Computational — Practice

Answer key — The Pythagorean theorem

1.
$c^2 = 8^2 + 15^2 = 64 + 225 = 289$ , so $c = \sqrt{289} = 17$ cm. $(8, 15, 17)$ is a Pythagorean triple.
2.
$5^2 + b^2 = 13^2$ gives $b^2 = 169 - 25 = 144$ , so $b = 12$ .
3.
The legs are $|7 - 1| = 6$ and $|-3 - 5| = 8$ , so the distance is $\sqrt{36 + 64} = \sqrt{100} = 10$ .
4.
Only $7, 24, 25$ : $7^2 + 24^2 = 49 + 576 = 625 = 25^2$ . The converse of the Pythagorean theorem then guarantees a right angle opposite the side $25$ . The others give $41 \ne 36$ , $85 \ne 81$ and $13 \ne 16$ .
5.
$c$ must be the hypotenuse, $10$ . So $6^2 + b^2 = 10^2$ , $b^2 = 64$ , $b = 8$ : the familiar $(6, 8, 10)$ triangle. Rahul's answer $11.7$ is longer than the hypotenuse, which can never happen.
6.
A square needs four equal sides and four right angles. The sides are hypotenuses, all of length $c$ . At each corner, the angle of the middle shape and the angles $\alpha$ , $\beta$ of two triangles fill a straight angle, so it is $180^\circ - (\alpha + \beta)$ . In a right triangle $\alpha + \beta = 180^\circ - 90^\circ = 90^\circ$ , so each corner is $90^\circ$ .
7.
$m^2 - n^2 = 21$ , $2mn = 20$ , $m^2 + n^2 = 29$ . Check: $21^2 + 20^2 = 441 + 400 = 841 = 29^2$ . ✓ Since $5$ and $2$ have no common factor and are not both odd, $(20, 21, 29)$ is primitive.
8.
$7^2 + 8^2 = 49 + 64 = 113 \lt 121 = 11^2$ , so the angle opposite $11$ is obtuse. (With legs $7$ and $8$ at a right angle, the hypotenuse would be $\sqrt{113} \approx 10.6$ ; a side of $11$ needs the angle to open wider.)
9.
The theorem says “if the angle at $C$ is right, then $AB^2 = CA^2 + CB^2$ ” (option b). Its converse swaps the two parts: “if $AB^2 = CA^2 + CB^2$ (and $A \ne C$ ), then the angle at $C$ is right” — option (a).
10.
```
def is_right(a, b, c):
    x, y, z = sorted([a, b, c])
    return x * x + y * y == z * z
```
The starter assumed that c is the hypotenuse — exactly the misconception of taking $c$ to be any side. Sorting makes the longest side play the role of $c$ , and by the converse the equation then decides whether there is a right angle.