# Intercept theorem examples

Example 1

A lady wants to get onto a flat roof and needs to work out what size ladder she needs. 1.8 metres up there is a bracket sticking out the wall. The bracket casts a shadow 3 metres away from the base. The flat roof casts a shadow 8 metres from the base. How high is the roof?

Redraw this diagram as:

Remember -  Thales intercept theorem where everything is proportional to the whole length of the triangles sides if there are parallel lines.

Therefore 3/8=1.8/x

Rearrange x to be the subject of the formula.

x=(1.8times8)/3

x=4.8m

Answer: The height of the flat roof is 4.8m.

Example 2

In the following diagram what is the value of x?

Remember that Thales intercept theorem (with parallel lines) everything is proportional to the whole length of the triangles sides.

Therefore 3:(3+6) is proportional to 2:x

Or

3/(3+6)=2/x

3/9=2/x

Rearrange to make x the subject of the formula:

x=(2times9)/3=18/3

x=6

NOTE:

3:6 is not proportional to 2:x because 6 is not the full length of the side of the triangle.

Example 3

In the above diagram AB is parallel to DC how long is ED?

Answer:

If AB and CD are parallel then these two triangles are similar and so the diagram above can be
re-drawn as:

Triangle ABE has been spun through 180° with point E as the pivot point.

Now the question how long is ED is easy.

Using Thales intercept theorem (with parallel lines) the triangle sides would be proportional i.e.

8.5/6=(ED)/4

Rearrange the formula so that ED is the subject of the formula.

ED=(8.5times4)/6

Answer: ED = 5.66

Example 4

Are lines AB – CD parallel?

If they were parallel then these two triangles are similar and so can be re-arranged as:

And using Thales intercept theorem (with parallel lines) full length triangle sides would be proportional i.e.

4/10 Would be proportional to 6/15

2/5=0.4         3/7.5=0.4

As they are the same answer Yes the lines AB – CD are parallel

Example 5

If the lines CD and BE are parallel which of the following equations are correct?

(AC)/(CD)=(BE)/(CD)=(AB)/(AE) or \color{#512d86}\((AB)/(AC)=(AE)/(AD)=(BE)/(CD)) or (CB)/(CA)=(DE)/(DA)=(CD)/(BE) or (CB)/(CA)=(DE)/(DA)=(BE)/(CD)

To check this out remember Thales intercept theorem (with parallel lines) everything is proportional to the whole length of the triangles sides.

Example 6

In the following diagram how long is AB?

Therefore 6/(AB)=4/7

AB=(6times7)/4=42/4=21/2=10.5

Answer: AB is 10.5

Example 7

Are the following lines AB and CD parallel?

If they were then 6/12=4/8

1/2=1/2 So yes AB and CD are parallel.