Isosceles triangle perimeter 30, base 8, find the equal sides Easy

The question

Question: find the two equal sides of an isosceles triangle.

The perimeter is 30.

The base is 8.

1 Use the isosceles fact

Isosceles means two sides are equal.

The small marks show which two.

We call each of them x.

2 Write the perimeter

The perimeter is all the sides added: the way round the edge.

So x + x + 8 = 30.

Careful!

30 take away 8 is both equal sides together.

One side is half of it.

3 Group like terms

Terms are the parts between the + and − signs.

Like terms are the same kind: same letters, same powers.

All the x terms go together.

x + x = 2x.

Now we have 2x + 8 = 30.

We want x by itself.

Two things are stuck to x: times 2 and + 8.

The + 8 is on the outside, so it comes off first.

Then the times 2.

4 Remove the + 8

To remove + 8, we take away 8.

We do it on both sides.

Why both sides?

The = sign means both sides are the same amount.

If we change only one side, they are not the same any more.

+ 8 and − 8 make zero.

Cross them out.

30 − 8 = 22.

Now we have 2x = 22.

5 Remove the 2

2x means 2 times x.

We want just x.

To remove times 2, we divide by 2.

We do it on both sides, so they stay equal.

2 divided by 2 is 1.

So just x is left.

22 divided by 2 is 11.

So x = 11!

So each equal side is 11.

6 Check

Let's check: put x = 11 back and add the three sides.

We get the perimeter, 30.

So x = 11 is right!

Let's look back.

First: use the isosceles fact.

Then: write the perimeter.

Then: group like terms.

Then: remove the + 8.

Then: remove the 2.

Last: check.

▶ Watch it solved

Isosceles triangle equal sides 2x + 3 and 3x - 4 Medium

The question

Question: find x and the length of the marked sides.

An isosceles triangle has sides 2x + 3 and 3x − 4.

1 Use the isosceles fact

Isosceles means two sides are equal.

The marks show which two.

So the two marked sides are equal: 2x + 3 = 3x − 4.

Careful!

The marks say the sides are the same length.

So we write one equals the other.

2 Get x on one side

There is x on both sides.

We want x on one side only.

So we remove the 2x from the left side.

To remove 2x, we take away 2x.

We do it on both sides.

Why both sides?

The = sign means both sides are the same amount.

If we change only one side, they are not the same any more.

2x and − 2x make zero.

Cross them out.

3x − 2x = x.

Now we have 3 = x − 4.

Let's swap the two sides, so x is on the left.

3 Remove the − 4

We want x by itself.

The − 4 is in the way.

To remove − 4, we add 4.

We do it on both sides, so they stay equal.

− 4 and + 4 make zero.

Cross them out.

3 + 4 = 7.

So x = 7!

4 Find the side

Put x = 7 in the side 2x + 3.

So each marked side is 17.

5 Check

Let's check: put x = 7 in the other side, 3x − 4.

We get 17 again: the sides are equal.

So x = 7 is right!

Let's look back.

First: use the isosceles fact.

Then: get x on one side.

Then: remove the − 4.

Then: find the side.

Last: check.

▶ Watch it solved

Equilateral triangle angle 2x + 10 Harder

The question

Question: find x.

One angle of an equilateral triangle is 2x + 10.

1 Use the equilateral fact

Equilateral means all three sides are equal.

The marks show it.

Equal sides make equal angles.

So all three angles are the same.

They add up to 180°.

180 shared by 3 is 60.

So every angle is 60°.

So 2x + 10 = 60.

Careful!

One angle is not the whole 180°.

It is one third of it: 60°.

We want x by itself.

Two things are stuck to x: times 2 and + 10.

The + 10 is on the outside, so it comes off first.

Then the times 2.

2 Remove the + 10

To remove + 10, we take away 10.

We do it on both sides.

Why both sides?

The = sign means both sides are the same amount.

If we change only one side, they are not the same any more.

+ 10 and − 10 make zero.

Cross them out.

60 − 10 = 50.

Now we have 2x = 50.

3 Remove the 2

2x means 2 times x.

We want just x.

To remove times 2, we divide by 2.

We do it on both sides, so they stay equal.

2 divided by 2 is 1.

So just x is left.

50 divided by 2 is 25.

So x = 25!

4 Check

Let's check: put x = 25 in the angle.

We get 60°.

So x = 25 is right!

Let's look back.

First: use the equilateral fact.

Then: remove the + 10.

Then: remove the 2.

Last: check.

▶ Watch it solved