Routes: the length of a route on a map and in a neighbourhood

The plane and space, grade 10 (3 units) Lesson 4 of 15Learning

Routes: the length of a route on a map and in a neighbourhood

A route and its length

A route is a way from place to place along roads, paths or streets. The length of a route is the sum of the lengths of its parts.

The map below shows five villages, $\displaystyle A$ to $\displaystyle E$ , and the roads between them. Beside each road is its length in km. The map is not to scale: the lengths are written, not measured with a ruler.

The shortest route

Yoav lives in $\displaystyle A$ and goes to school in $\displaystyle E$ . The example below finds the shortest route from $\displaystyle A$ to $\displaystyle E$ by going over every route that does not pass the same village twice.

$\displaystyle A\to B\to E$ : $\displaystyle {12+9}={21}$ km.
$\displaystyle A\to B\to D\to E$ : $\displaystyle {12+4+6}={22}$ km.
$\displaystyle A\to C\to D\to E$ : $\displaystyle {7+5+6}={18}$ km.
$\displaystyle A\to C\to D\to B\to E$ : $\displaystyle {7+5+4+9}={25}$ km.
The shortest: $\displaystyle A\to C\to D\to E$ , 18 km – although it takes 3 roads, and $\displaystyle A\to B\to E$ only 2.

Exercise 1

Use the map above. How many km?

The route

\displaystyle E\to D\to C\to A

/ – 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

The shortest route from

\displaystyle B

\displaystyle C

/ – 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

The shortest route from

\displaystyle A

\displaystyle D

/ – 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 route in the streets of a neighbourhood

In the neighbourhood below the streets are straight and all meet at right angles – like the lines of squared paper. The distance between two neighbouring crossings is 100 m.
Noa walks from her home, $\displaystyle A$ , to a friend who lives at $\displaystyle B$ .

$\displaystyle B$ is 4 streets to the right of $\displaystyle A$ and 3 streets up.
Every route must go 4 parts to the right and 3 up in all: $\displaystyle {4+3}={7}$ parts, $\displaystyle {7\cdot100}={700}$ m.
There are many such routes – the route in the drawing, $\displaystyle A\to P\to Q\to B$ , is only one of them – and all have the same length, 700 m.
A route that also goes left or down is longer: each such part has to be made up for in the opposite direction.

Exercise 2

In a neighbourhood, the distance between two neighbouring crossings is 50 m.
Uri walked from $\displaystyle A$ to $\displaystyle B$ along the route $\displaystyle A\to P\to Q\to B$ .

The length of Uri’s route, 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

The length of the shortest route from

\displaystyle A

\displaystyle B

along the streets, 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

How many metres longer is Uri’s route?

/ – 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)

Open as a page

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
$\displaystyle \alpha_1+\alpha_2+\ldots=360^{\circ}$