Worked examples
The question
Question: can sides of 3, 4 and 8 make a triangle?
sides 3, 4, 8
For a triangle, the two short sides together must be longer than the longest side.
Else they cannot meet.
the two short sides must reach
round the longest one
1 Add the two short sides
The longest side is 8.
We lay it down first.
The two short sides are 3 and 4.
Together: 7.
= 3 + 4 = 7
2 Compare with the longest
Swing them up from the ends.
7 is less than 8: they do not even reach each other.
Careful!
Adding the long side to a short one is always enough.
The test is the two short sides.
Careful: add the two SHORT sides
So no: they cannot make a triangle.
= 7 < 8 → no triangle
3 Check
Let's check the other two pairs: 3 plus 8 is 11, and 4 plus 8 is 12.
Both are more than the third side.
So only the short pair decides.
So the answer is right!
3 + 8 = 11 > 4, 4 + 8 = 12 > 3
Let's look back.
First: add the two short sides.
Then: compare with the longest.
Last: check.
Answer = 7 < 8 → no triangle
▶ Watch it solved
The question
Question: what can the third side be, if a triangle has sides 5 and 9?
sides 5 and 9, third side = ?
In a triangle, any two sides together are longer than the third side.
any two sides together
are longer than the third
1 Longest it can be
The third side must be shorter than 5 plus 9, which is 14.
Else the two sides cannot reach round it.
= third < 5 + 9 = 14
2 Shortest it can be
And the third side plus 5 must be longer than 9.
So it must be longer than 9 minus 5: 4.
= third > 9 − 5 = 4
Careful!
The third side can be the longest side.
It only has to be shorter than 14.
Careful: not third < 9 ✗
So the third side is between 4 and 14, but not 4 or 14 itself.
= 4 < third side < 14
3 Check
Let's check with a third side of 10, between 4 and 14: 5 plus 9 is 14, more than 10.
5 plus 10 is 15, more than 9.
A triangle works.
So the answer is right!
10: 5 + 9 = 14 > 10, 5 + 10 = 15 > 9
Let's look back.
First: longest it can be.
Then: shortest it can be.
Last: check.
Answer = 4 < third side < 14
▶ Watch it solved
The question
Question: can sides of 8, 15 and 7 make a triangle?
sides 8, 15, 7
For a triangle, the two short sides together must be longer than the longest side.
Else they cannot meet.
the two short sides must reach
round the longest one
1 Add the two short sides
The longest side is 15.
We lay it down first.
The two short sides are 7 and 8.
Together: 15.
= 7 + 8 = 15
2 Compare with the longest
Swing them up from the ends.
15 is exactly 15: they only meet lying flat on the long side.
Careful!
7 plus 8 is exactly 15.
The sides lie flat on top of each other.
That is a line, not a triangle.
Careful: equal is not enough: it lies flat
So no: they cannot make a triangle.
= 15 = 15 → no triangle
3 Check
Let's check the other two pairs: 7 plus 15 is 22, and 8 plus 15 is 23.
Both are more than the third side.
So only the short pair decides.
So the answer is right!
7 + 15 = 22 > 8, 8 + 15 = 23 > 7
Let's look back.
First: add the two short sides.
Then: compare with the longest.
Last: check.
Answer = 15 = 15 → no triangle
▶ Watch it solved
