Practice: routes

The plane and space, grade 10 (3 units) Lesson 7 of 15Practice

Practice: routes

1

The map shows five villages and the roads between them. Beside each road is its length in km.
How many km?

a

The route

\displaystyle A\to B\to C\to E

/ – 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

b

The shortest route from

\displaystyle A

to

\displaystyle E

/ – 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

c

The shortest route from

\displaystyle B

to

\displaystyle E

/ – 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

2

In a neighbourhood whose streets make a grid, the distance between two neighbouring crossings is 80 m.
The library is 5 streets east and 2 streets north of Amit’s home.
How long is the shortest route along the streets, in metres?

/ – 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

3

The measurements in the drawing are in metres.
How long is the diagonal path $\displaystyle AC$ across the rectangular plot?

/ – 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

4

Dana runs 5 km in 25 minutes.
What is her speed, in km/h?

/ – 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

5

From the city to the port you can go by bus or by train.
The bus: 40 km, at an average speed of 60 km/h.
The train: 50 km, at an average speed of 100 km/h, and before the journey you wait 10 minutes for it.

a

The time by bus, in minutes

/ – 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

b

The time by train, with the wait, in minutes

/ – 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

c

Which is worth choosing? Explain.

6

On a squared map, each square is 5 m by 5 m.
$\displaystyle B$ is 9 squares to the right of $\displaystyle A$ and 12 squares up.

a

The 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 – 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

b

The way in a straight line, in metres

/ – 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

c

By how many metres is the straight line shorter?

/ – 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

7

A train leaves at 8:15 for a 120 km journey, at an average speed of 80 km/h.
At what time does it arrive?

8

A diagonal path 26 m long crosses a rectangular yard 10 m wide.
How long is the yard, in metres?

/ – 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

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

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