# 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.

# 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.