Pythagorean theorem and irrational numbers

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1. Square roots and cube roots

Warm-up

Three quick questions from earlier lessons.

Discover

A square tile has an area of 49 cm². How long is a side?

7 cm
24.5 cm

Key idea

Square roots undo squaring; cube roots undo cubing.

\sqrt{a}

is the positive number whose square is

a

(the side of a square of area

a

\sqrt[3]{a}

is the number whose cube is

a

(the edge of a cube of volume

a

). If

a

is not a perfect square,

\sqrt{a}

lies between two consecutive whole numbers.

Worked example

$\sqrt{30}$ and $\sqrt[3]{64}$ .

5 < \sqrt{30} < 6

25 < 30 < 36.

\sqrt[3]{64} = 4

4 · 4 · 4 = 64.

Your turn, guided

Your turn: the edge of a cube of volume 125.

5

5³ = 125.

Practice ★

A square has an area of 225 square units. How long is its side?
A square has an area of 144 square units. How long is its side?

A cube has a volume of 343 cubic units. How long is each edge?

Watch out!

Practice ★★

$\sqrt{ 37 }$ lies between two consecutive whole numbers. What is the smaller one?
$\sqrt{ 17 }$ lies between two consecutive whole numbers. What is the smaller one?

Exit ticket

Three questions to check you have got it.

A square has an area of 225 square units. How long is its side?
A square has an area of 144 square units. How long is its side?
A square has an area of 49 square units. How long is its side?

Summary card

Square and cube roots

$\sqrt{a}$ : side of a square of area $a$ ; $\sqrt[3]{a}$ : edge of a cube of volume $a$ .

Watch out: a root is not half or a third.

2. Rational and irrational numbers; repeating decimals

Warm-up

Three quick questions from earlier lessons.

Discover

$\tfrac{1}{3} = 0.333\ldots$ goes on forever. Is it rational?

Yes
No: it never ends

Key idea

Rational numbers are fractions; their decimals end or repeat.

A rational number can be written

\tfrac{a}{b}

with integers

a

b \neq 0

; its decimal expansion terminates or repeats. An irrational number cannot: its decimal never ends and never repeats.

\sqrt{n}

is irrational unless

n

is a perfect square;

\pi

is irrational.

Worked example

Write $0.\overline{27}$ as a fraction.

100x = 27.2727\ldots

Let x = 0.2727…

99x = 27

Subtract x.

x = \tfrac{27}{99} = \tfrac{3}{11}

Simplify.

Your turn, guided

Your turn: write $0.\overline{5}$ as a fraction.

\tfrac{5}{9}

10x − x = 5.

Practice ★

Which number is irrational?
1. $\sqrt{ 2 }$
2. $\sqrt{ 4 }$
3. $0.\overline{ 1 }$
Which number is irrational?
1. $\sqrt{ 10 }$
2. $\sqrt{ 16 }$
3. $0.\overline{ 3 }$

Write $0.\overline{ 1 }$ (the digit 1 repeating forever) as a fraction.

Watch out!

Practice ★★

Write $0.\overline{ 48 }$ (the digits 48 repeating forever) as a fraction.
Write $0.\overline{ 70 }$ (the digits 70 repeating forever) as a fraction.

Exit ticket

Three questions to check you have got it.

Write $0.\overline{ 1 }$ (the digit 1 repeating forever) as a fraction.
Write $0.\overline{ 2 }$ (the digit 2 repeating forever) as a fraction.
Write $0.\overline{ 8 }$ (the digit 8 repeating forever) as a fraction.

Summary card

Rational and irrational

Ends or repeats: rational. Neither: irrational.

Watch out: a repeating decimal is a fraction over 9, 99, …

3. The Pythagorean theorem

Warm-up

Three quick questions from earlier lessons.

Discover

A right triangle has legs 3 and 4. Squares are drawn on all three sides. The squares on the legs have areas 9 and 16. What is the area of the square on the hypotenuse?

Set the legs to 3 and 4, then try other lengths. Which rule does the hypotenuse follow?

The hypotenuse is √(a² + b²)
The hypotenuse is a + b

Key idea

The Pythagorean theorem.

In a right triangle with legs

a

b

and hypotenuse

c

(the side opposite the right angle, the longest side):

a^2 + b^2 = c^2

. To find a leg, subtract:

b^2 = c^2 - a^2

Worked example

Hypotenuse 10, one leg 6.

b^2 = 100 - 36 = 64

Subtract the squares.

b = 8

Square root.

Your turn, guided

Your turn: legs 8 and 15. The hypotenuse.

17

√(64 + 225) = √289.

Practice ★

A right triangle has legs 10 and 24. How long is the hypotenuse?
A right triangle has legs 20 and 48. How long is the hypotenuse?

A right triangle has legs 2 and 4. Find the exact length of the hypotenuse.

Watch out!

Practice ★★

A right triangle has hypotenuse 15 and one leg 9. How long is the other leg?
A right triangle has hypotenuse 25 and one leg 15. How long is the other leg?

Exit ticket

Three questions to check you have got it.

A right triangle has legs 10 and 24. How long is the hypotenuse?
A right triangle has legs 20 and 48. How long is the hypotenuse?
A right triangle has legs 15 and 36. How long is the hypotenuse?

Summary card

Pythagorean theorem

$a^2 + b^2 = c^2$ , with $c$ the hypotenuse.

Watch out: for a leg, subtract the squares.

4. The converse of the Pythagorean theorem; applications and distances

Warm-up

Three quick questions from earlier lessons.

Discover

A triangle has sides 6, 8 and 10. Is it a right triangle?

Yes
You cannot tell

Key idea

The converse, and right triangles everywhere.

Converse: if

a^2 + b^2 = c^2

for the two shorter sides

a

b

and the longest side

c

, the triangle is a right triangle. Use the theorem for ladders, diagonals and distances between points: the horizontal and vertical changes are the legs.

Worked example

Distance from $(1, 2)$ to $(7, 10)$ .

Legs 6 and 8

Changes.

Distance 10

√(36 + 64).

Your turn, guided

Your turn: distance from $(0, 0)$ to $(5, 12)$ .

13

√(25 + 144).

Practice ★

Is a triangle with sides 3, 4 and 5 a right triangle?
1. Yes: the squares of the two shorter sides add up to the square of the longest
2. No: the sides are not consecutive whole numbers
3. You cannot tell without measuring an angle
Is a triangle with sides 9, 12 and 15 a right triangle?
1. Yes: the squares of the two shorter sides add up to the square of the longest
2. No: the sides are not consecutive whole numbers
3. You cannot tell without measuring an angle

A 13 ft ladder leans against a wall and reaches 12 ft up. How far is its foot from the wall?

Watch out!

Practice ★★

Is a triangle with sides 7, 8 and 9 a right triangle?
1. No: 49 + 64 is not 81
2. Yes: like 3, 4, 5, the sides are consecutive
3. You cannot tell without measuring an angle
Is a triangle with sides 6, 7 and 8 a right triangle?
1. No: 36 + 49 is not 64
2. Yes: like 3, 4, 5, the sides are consecutive
3. You cannot tell without measuring an angle

Boss problem. Find the distance between $A(-1, -5)$ and $B(5, 3)$ using a right triangle.

Exit ticket

Three questions to check you have got it.

A 13 ft ladder leans against a wall and reaches 12 ft up. How far is its foot from the wall?
A 39 ft ladder leans against a wall and reaches 36 ft up. How far is its foot from the wall?
A 26 ft ladder leans against a wall and reaches 24 ft up. How far is its foot from the wall?

Summary card

Converse and applications

$a^2 + b^2 = c^2$ means right angle; distances are hypotenuses.

Watch out: always test with the longest side as $c$ .

5. Distance and midpoint in the coordinate plane

Warm-up

Three quick questions from earlier lessons.

Discover

From $(1, 1)$ to $(4, 5)$ you move 3 right and 4 up. How long is the straight segment?

Put A at (0, 0) and B at (4, 6). Where is M?

(2, 3)
(4, 6)

Key idea

Pythagoras on the grid, and averaging for the middle.

Distance:

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

. Midpoint:

\left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right)

Worked example

$A(-2, 3)$ , $B(4, -5)$ .

d = \sqrt{36 + 64} = 10

Δx = 6, Δy = −8.

Midpoint

(1, -1)

Average the coordinates.

Your turn, guided

Your turn: the distance from $(0, 0)$ to $(5, 12)$ .

13

√(25 + 144).

Practice ★

Find the distance between $(-1, -3)$ and $(2, 1)$ .
Find the distance between $(-3, 3)$ and $(0, 7)$ .

Tap the midpoint of the segment from $(-6, -5)$ to $(-4, -1)$ .
Tap the midpoint of the segment from $(-7, -8)$ to $(1, -4)$ .

Watch out!

Practice ★★

Find the exact distance between $(0, -2)$ and $(1, -4)$ .
Find the exact distance between $(2, -2)$ and $(3, -6)$ .

Exit ticket

Three questions to check you have got it.

Find the distance between $(-1, -3)$ and $(2, 1)$ .
Find the distance between $(-3, 3)$ and $(0, 7)$ .
Find the distance between $(2, 5)$ and $(11, 17)$ .

Summary card

Distance and midpoint

$d = \sqrt{\Delta x^2 + \Delta y^2}$ ; midpoint = average of the coordinates.

Watch out: average, not half the difference.

Answers: Pythagorean theorem and irrational numbers

1. Square roots and cube roots

$15$
$12$

$15$
$12$
$7$

2. Rational and irrational numbers; repeating decimals

$\sqrt{ 2 }$
$\sqrt{ 10 }$

$\frac{1}{9}$

$\frac{16}{33}$
$\frac{70}{99}$

$\frac{1}{9}$
$\frac{2}{9}$
$\frac{8}{9}$

3. The Pythagorean theorem

$26$
$52$

$2 \sqrt{5}$

$12$
$20$

$26$
$52$
$39$

4. The converse of the Pythagorean theorem; applications and distances

Yes: the squares of the two shorter sides add up to the square of the longest
Yes: the squares of the two shorter sides add up to the square of the longest

No: 49 + 64 is not 81
No: 36 + 49 is not 64

$10$

$5$
$15$
$10$

5. Distance and midpoint in the coordinate plane

$(-5, -3)$
$(-3, -6)$

$\sqrt{5}$
$\sqrt{17}$

$5$
$5$
$15$