DE ∥ BC, AD = 4, DB = 6, AE = 6, find EC Easy

The question

Question: D E is parallel to B C, with AD = 4, DB = 6, AE = 6: find EC.

1 Use the parallel line

D E is parallel to B C: the arrows show it.

So triangle A D E has the same angles as A B C: they are similar.

A parallel line cuts the two sides in the same ratio: A D to D B is the same as A E to E C.

2 Write the proportion

So: AD over DB equals AE over EC.

Careful!

Keep the same order on both sides: the matching parts on top, the matching parts at the bottom.

Put the numbers in.

We call EC x: 4/6 = 6 over x.

3 Cross multiply

When two fractions are equal, we can cross multiply.

These two are equal.

Work out the multiplications.

4 Remove the 4

4x means 4 times x.

We want just x.

To remove times 4, we divide by 4.

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.

4 divided by 4 is 1.

So just x is left.

36 divided by 4 is 9.

So x = 9!

5 Check

Let's check: put 9 in place of x.

First the left side.

Now the right side.

Both sides are 2/3.

So x = 9 is right!

So EC = 9.

Let's look back.

First: use the parallel line.

Then: write the proportion.

Then: cross multiply.

Then: remove the 4.

Last: check.

▶ Watch it solved

DE ∥ BC, AD = 6, DB = 4, AE = 9, find EC Medium

The question

Question: D E is parallel to B C, with AD = 6, DB = 4, AE = 9: find EC.

1 Use the parallel line

D E is parallel to B C: the arrows show it.

So triangle A D E has the same angles as A B C: they are similar.

A parallel line cuts the two sides in the same ratio: A D to D B is the same as A E to E C.

2 Write the proportion

So: AD over DB equals AE over EC.

Careful!

Keep the same order on both sides: the matching parts on top, the matching parts at the bottom.

Put the numbers in.

We call EC x: 6/4 = 9 over x.

3 Cross multiply

When two fractions are equal, we can cross multiply.

These two are equal.

Work out the multiplications.

4 Remove the 6

6x means 6 times x.

We want just x.

To remove times 6, we divide by 6.

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.

6 divided by 6 is 1.

So just x is left.

36 divided by 6 is 6.

So x = 6!

5 Check

Let's check: put 6 in place of x.

First the left side.

Now the right side.

Both sides are 3/2.

So x = 6 is right!

So EC = 6.

Let's look back.

First: use the parallel line.

Then: write the proportion.

Then: cross multiply.

Then: remove the 6.

Last: check.

▶ Watch it solved

DE ∥ BC, AD = 7, DB = 14, EC = 12, find AE Harder

The question

Question: D E is parallel to B C, with AD = 7, DB = 14, EC = 12: find AE.

1 Use the parallel line

D E is parallel to B C: the arrows show it.

So triangle A D E has the same angles as A B C: they are similar.

A parallel line cuts the two sides in the same ratio: A D to D B is the same as A E to E C.

2 Write the proportion

So: AD over DB equals AE over EC.

Careful!

Keep the same order on both sides: the matching parts on top, the matching parts at the bottom.

Put the numbers in.

We call AE x: 7/14 = x/12.

3 Cross multiply

When two fractions are equal, we can cross multiply.

These two are equal.

Work out the multiplications.

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

4 Remove the 14

14x means 14 times x.

We want just x.

To remove times 14, we divide by 14.

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.

14 divided by 14 is 1.

So just x is left.

84 divided by 14 is 6.

So x = 6!

5 Check

Let's check: put 6 in place of x.

First the left side.

Now the right side.

Both sides are 1/2.

So x = 6 is right!

So AE = 6.

Let's look back.

First: use the parallel line.

Then: write the proportion.

Then: cross multiply.

Then: remove the 14.

Last: check.

▶ Watch it solved