Practice: The Pythagorean theorem

The Pythagorean theorem

Warm-upHuman

A right triangle has legs $8$ cm and $15$ cm. How long is the hypotenuse, in cm?

The Pythagorean theorem

Warm-upHuman

A right triangle has hypotenuse $13$ and one leg $5$ . How long is the other leg?

The Pythagorean theorem

CoreHuman

What is the straight-line distance between the points $(1, 5)$ and $(7, -3)$ ?

The Pythagorean theorem

CoreHuman

Which of these triangles has a right angle?

Choose one

Sides

7

,

24

,

25

Sides

4

,

5

,

6

Sides

6

,

7

,

9

Sides

2

,

3

,

4

The Pythagorean theorem

CoreHuman

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?

Choose one

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

.Nothing — the third side is

\sqrt{136}

.He should have added the lengths: the third side is

16

.The theorem only works for whole-number triangles.

The Pythagorean theorem

CoreHuman

In the first rearrangement proof, why is the shape in the middle of the left picture a square?

Choose one

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.Because its four sides are all equal to

c

.Because it looks like a square in the picture.Because the big outer shape is a square.

The Pythagorean theorem

StretchHuman

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?

The Pythagorean theorem

StretchHuman

A triangle has sides $7$ , $8$ and $11$ . What kind of angle is opposite the side $11$ ?

Choose one

Acute (less than

90^\circ

)A right angleObtuse (more than

90^\circ

)

The Pythagorean theorem

CoreFormal

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?

Choose one

A ≠ C → sqDist A B = sqDist C A + sqDist C B → IsRightAngleAt A B CIsRightAngleAt A B C → sqDist A B = sqDist C A + sqDist C BsqDist A B = sqDist C A + sqDist C BIsRightAngleAt A B C → sqDist A B = sqDist C A * sqDist C B

The Pythagorean theorem

StretchComputational

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.

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