A shortcut: Pythagoras' theorem

The plane and space, grade 10 (3 units) Lesson 5 of 15Learning

A shortcut: Pythagoras' theorem

Pythagoras’ theorem

In a right triangle, the two sides that make the right angle are the legs, $\displaystyle a$ and $\displaystyle b$ , and the side opposite the right angle is the hypotenuse, $\displaystyle c$ – the longest side.

\displaystyle c^2

\displaystyle =

\displaystyle a^2+b^2

So the hypotenuse is the square root of the sum of the squares:

\displaystyle c

\displaystyle =

\displaystyle \sqrt{a^2+b^2}

Example: crossing a park diagonally

A park is the rectangle $\displaystyle ABCD$ , 80 m by 60 m.
Shira has to get from the corner $\displaystyle A$ to the corner $\displaystyle C$ .
The example below compares walking along two sides with walking on the diagonal path $\displaystyle AC$ .

Along the sides, $\displaystyle A\to B\to C$ : $\displaystyle {80+60}={140}$ m.
The triangle $\displaystyle ABC$ has a right angle at $\displaystyle B$ , and $\displaystyle AC$ is the hypotenuse: $\displaystyle {AC}={\sqrt{80^2+60^2}}={\sqrt{6400+3600}}={\sqrt{10000}}={100}$ m.
On the diagonal path Shira saves $\displaystyle {140-100}={40}$ m.
A straight line is always the shortest way between two points; Pythagoras tells by how much.

Exercise 1

Calculate the length of the diagonal path, in metres.

A rectangular lawn, 120 m by 50 m

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

A rectangular yard, 12 m by 9 m

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

A football pitch, 100 m by 75 m

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

A straight line on squared paper

On squared paper, a straight line from $\displaystyle A$ to $\displaystyle B$ is the hypotenuse of a right triangle whose legs are on the lines of the squares.
Here $\displaystyle B$ is 4 squares to the right of $\displaystyle A$ and 3 squares up.

Along the lines of the squares: $\displaystyle {4+3}={7}$ squares.
In a straight line: $\displaystyle {\sqrt{4^2+3^2}}={\sqrt{25}}={5}$ squares.

Exercise 2

A cyclist left a village, rode 9 km east, and then 12 km north.
The roads meet at a right angle.

How long is a straight road from the village to where he got to, in km?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

By how many km is the straight road shorter?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

A map has a grid of squares drawn on it, and each square is 10 m by 10 m.
The spring is 6 squares to the right of the car park and 8 squares up.

A way along the lines of the squares, in metres

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

A way in a straight line, in metres

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Finding a leg

When the hypotenuse and one leg are known, the other leg is found by taking the square of the known leg from the square of the hypotenuse.

\displaystyle a

\displaystyle =

\displaystyle \sqrt{c^2-b^2}

Example. A path 13 m long crosses a rectangular yard 5 m wide diagonally.
The length of the yard: $\displaystyle {\sqrt{13^2-5^2}}={\sqrt{169-25}}={\sqrt{144}}={12}$ m.

Exercise 4

A diagonal path 25 m long crosses a rectangular plot 7 m wide.
Find the length of the plot, in metres.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: the plane and space, grade 10 (3 units)

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Perimeters

The perimeter of a rectangle – $\displaystyle a$ , $\displaystyle b$ two sides that meet at a vertex
$\displaystyle P$ $\displaystyle =$ $\displaystyle 2(a+b)$
The perimeter of a square – $\displaystyle a$ the side
$\displaystyle P$ $\displaystyle =$ $\displaystyle 4a$
Pi
$\displaystyle \pi\approx3.14$
The circumference – $\displaystyle d$ the diameter
$\displaystyle C$ $\displaystyle =$ $\displaystyle \pi d$
The circumference – $\displaystyle r$ the radius
$\displaystyle C$ $\displaystyle =$ $\displaystyle 2\pi r$

Routes: Pythagoras, distance, speed and time

Pythagoras’ theorem – $\displaystyle a$ , $\displaystyle b$ the legs, $\displaystyle c$ the hypotenuse
$\displaystyle c^2$ $\displaystyle =$ $\displaystyle a^2+b^2$
The hypotenuse
$\displaystyle c$ $\displaystyle =$ $\displaystyle \sqrt{a^2+b^2}$
A leg
$\displaystyle a$ $\displaystyle =$ $\displaystyle \sqrt{c^2-b^2}$
Speed – $\displaystyle s$ the distance, $\displaystyle t$ the time
$\displaystyle v$ $\displaystyle =$ $\displaystyle \frac{s}{t}$
Distance
$\displaystyle s$ $\displaystyle =$ $\displaystyle v\cdot t$
Time
$\displaystyle t$ $\displaystyle =$ $\displaystyle \frac{s}{v}$

Areas

The area of a rectangle
$\displaystyle A$ $\displaystyle =$ $\displaystyle a\cdot b$
The area of a square
$\displaystyle A$ $\displaystyle =$ $\displaystyle a^2$
The area of a triangle – $\displaystyle h_a$ the height to the side $\displaystyle a$
$\displaystyle A$ $\displaystyle =$ $\displaystyle \frac{a\cdot h_a}{2}$
The area of a parallelogram – $\displaystyle h_a$ the height to the side $\displaystyle a$
$\displaystyle A$ $\displaystyle =$ $\displaystyle a\cdot h_a$
The area of a trapezoid – $\displaystyle a$ , $\displaystyle b$ the bases, $\displaystyle h$ the height
$\displaystyle A$ $\displaystyle =$ $\displaystyle \frac{(a+b)\cdot h}{2}$
The area of a disc – $\displaystyle r$ the radius
$\displaystyle A$ $\displaystyle =$ $\displaystyle \pi r^2$
A square metre in square centimetres (a dunam is 1000 m²)
$\displaystyle 1\text{ m}^2=100\cdot100\text{ cm}^2=10000\text{ cm}^2$

Tilings

The sum of the angles of a polygon with $\displaystyle n$ sides
$\displaystyle 180^{\circ}\cdot(n-2)$
The angle of a regular polygon with $\displaystyle n$ sides
$\displaystyle \alpha$ $\displaystyle =$ $\displaystyle \frac{180^{\circ}\cdot(n-2)}{n}$
The angles around a vertex of a tiling
$\displaystyle \alpha_1+\alpha_2+\ldots=360^{\circ}$