Year 10MeasurementMeasuringYear 10 Maths

Pythagoras, distance, speed and time

What this lesson covers

For right-angled triangles, Pythagoras’ theorem states that the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides.
If (a,b,c) is a Pythagorean triple, then so is (ka,kb,kc), where k is a positive integer.
There is a fixed relationship between speed, distance, and time: speed = distance/time.
In position-time graphs, the gradient represents speed.
Decimal measures are used for very small durations (e.g. milliseconds); the rest of time measurement uses a different system, based principally on 12 and 60.

Key idea

One theorem, three different jobs

Right-angled triangles have a rule nothing else does.
It finds a missing side, and it tests whether a corner is square.
And it measures the distance between any two points on a grid.

Key idea

Name the sides first

The hypotenuse is the side opposite the right angle.
It is always the longest side. That is a useful check.
In a squared plus b squared equals c squared, c is that side.

Worked example

Finding a missing side

Short sides three and four: nine plus sixteen is twenty-five.
That is c squared, so c is five.
If the hypotenuse is known instead, subtract rather than add.

Given	Working	Answer
Sides 3 and 4	9 + 16	25
So c²= 25	square root	c = 5
Hypotenuse 13, side 5	169 − 25	144
Missing side	square root	12

Key idea

Add or subtract?

Looking for the longest side: add the two squares.
Looking for a short side: subtract from the longest squared.
If your answer is longer than the hypotenuse, you added by mistake.

Worked example

Is that corner actually square?

Sides six, eight and eleven. Square the two short ones and add.
A hundred. But eleven squared is a hundred and twenty-one.
They do not match, so there is no right angle in it.

Step	Working	Result
Sides 6, 8 and 11	test it
Short sides	36 + 64	100
Longest squared	11²	121
100 is not 121	so no right angle

Worked example

Triples, and why they multiply

Three, four, five works. So does six, eight, ten.
Scale all three sides and the relationship survives.
Recognising them saves you the arithmetic entirely.

Triple	Where from	Check
3, 4, 5	the famous one	9 + 16 = 25
6, 8, 10	that doubled	still works
9, 12, 15	tripled	still works
5, 12, 13	a different one	25 + 144 = 169

Worked example

Distance between two points

Across four and up three. Draw those in and look.
You have made a right-angled triangle without meaning to.
So the distance is the hypotenuse. Sixteen plus nine, root, five.

Key idea

That is the distance formula

The x difference and the y difference are the two short sides.
So the formula is Pythagoras with the gaps written out.
Nothing new to memorise. Draw the triangle instead.

Worked example

A four metre ladder

Base a metre and a half from the wall. How high does it reach?
Sixteen take two point two five is thirteen point seven five.
Root of that, three point seven one metres. Not a tidy number.

Step	Working	Result
Ladder 4 m	base 1.5 m out
So	16 − 2.25	13.75
Height reached	√13.75	3.71 m
Most answers	are not whole numbers

Key idea

Speed, and where it hides on a graph

Speed is distance divided by time. You know that already.
On a position-time graph, distance is up and time is across.
So distance over time is rise over run. The gradient is the speed.

Worked example

Reading the shape before the numbers

A steeper section means more distance in the same time. Faster.
A flat section means time passing and distance not changing.
That is not missing data. That is standing still.

On the graph	Means	So
A flat section	time passes	distance does not change
Which means	stopped
A steeper section	more distance per second	faster
Read the shape	before the numbers

Worked example

Working it out

A hundred and twenty metres, over twenty seconds.
Rise over run: a hundred and twenty divided by twenty.
Six metres per second.

Reading	Working	Answer
Distance up the side	time along the bottom
Gradient	distance ÷ time	which is speed
120 m in 20 s	120 ÷ 20	6 m/s
Steeper line	faster

Key idea

Time is the one thing that is not decimal

Sixty seconds, sixty minutes, twenty-four hours, twelve on the clock.
So two point five hours is two hours thirty, not two hours fifty.
Below a second it goes decimal again. Milliseconds behave normally.

Written	Working	Really
2.5 hours	half an hour is 30 min	2 h 30
Not 2 h 50	that is the trap
3.75 hours	three quarters is 45	3 h 45
90 minutes	90 ÷ 60	1.5 hours

Summary

What to take away

Seven lines, and the first three are one theorem.


a² + b² = c²	c is always the hypotenuse
Forwards	two sides give the third
Backwards	three sides test for a right angle
On a grid	across and up are the short sides
A triple scaled up	is still a triple
Position-time gradient	is the speed
2.5 hours	is 2 h 30, not 2 h 50

Your turn

Your turn: four questions

Pause here and work all four.
Questions 1 and 3 are both triples in disguise.
Question 2 wants a yes or no, with a reason.

	Question
1	Legs 9 and 12. Find the hypotenuse.
2	Is 7, 24, 25 a right-angled triangle?
3	Distance from (2, 1) to (7, 13)?
4	150 m in 12 s. Speed to 3 s.f.?

Show answer

1. 15. Eighty-one plus a hundred and forty-four is two hundred and twenty-five, and the square root of that is fifteen. It is the three, four, five triangle tripled. 2. Yes. Forty-nine plus five hundred and seventy-six is six hundred and twenty-five, which is exactly twenty-five squared. 3. 13. Across is five and up is twelve, so it is the five, twelve, thirteen triple. Twenty-five plus a hundred and forty-four is a hundred and sixty-nine. 4. 12.5 metres per second. Distance over time, and both numbers have at least three significant figures, so the answer keeps three.

Worked example

The answers

Fifteen. It is three, four, five tripled.
Yes: forty-nine plus five seven six is six two five, which is 25².
Thirteen. Across five, up twelve.
Twelve point five metres per second.

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