# Midpoint theorem examples

**Example 1**

Find the distance A to C

**Answer follows:**

The lines B to C and B to A are bisected at their midpoints

`x` to `y` must therefore be parallel to A to C

And midpoint theorem tells us that

`x` to `y` is `1/2 A` to `C`

Therefore

`A` to `C=2times6`

`A` to `C=12`

**Example 2**

Find the distance AB

**Answer follows:**

The lines `xy` and `xz` are bisected at their midpoints

Therefore `A` to `B` and `y` to `z` must be parallel

And midpoint theorem tells us that

`A` to `B` is `1/2yz`

Therefore

`A` to `B=5`

**Example 3**

Find the distance `x` to `y` in the following diagram

Solution

Using Pythagoras theorem `8^2+6^2=H^2`

Therefore

`H^2=64+36`

`H^2=100`

`H=10`

The lines `A` to `C` and `C` to `B` are bisected at their midpoints

`x` to `y` must therefore be parallel to `A` to `B`

And midpoints theorem tells us that

`x` to `y` is `1/2A` to `B`

Therefore `x` to `y` is `1/2times 10`

**Answer:** `x` to `y=5`

# Midpoint theorem examples

**Example 1**

Find the distance A to C

**Answer follows:**

The lines B to C and B to A are bisected at their midpoints

`x` to `y` must therefore be parallel to A to C

And midpoint theorem tells us that

`x` to `y` is `1/2 A` to `C`

Therefore

`A` to `C=2times6`

`A` to `C=12`

**Example 2**

Find the distance AB

**Answer follows:**

The lines `xy` and `xz` are bisected at their midpoints

Therefore `A` to `B` and `y` to `z` must be parallel

And midpoint theorem tells us that

`A` to `B` is `1/2yz`

Therefore

`A` to `B=5`

**Example 3**

Find the distance `x` to `y` in the following diagram

Solution

Using Pythagoras theorem `8^2+6^2=H^2`

Therefore

`H^2=64+36`

`H^2=100`

`H=10`

The lines `A` to `C` and `C` to `B` are bisected at their midpoints

`x` to `y` must therefore be parallel to `A` to `B`

And midpoints theorem tells us that

`x` to `y` is `1/2A` to `B`

Therefore `x` to `y` is `1/2times 10`

**Answer:** `x` to `y=5`