# Applying Pythagoras’ Theorem in 2-D

**Finding the lengths of diagonals**

Pythagoras’ Theorem can be used it to find the length of the diagonal of a rectangle given the lengths of the sides.

For example, use Pythagoras Theorem to calculate the length go *d*

*d²* = 10.2² + 13.6²

*d²* = 104.04 + 184.96

*d²* = 289

*d* = √289

*d* = 17 cm

**Finding the height of an isosceles triangle**

The Pythagoras Theorem can also be used to calculate the height of an isosceles triangle. For example:

Calculate the height of *h* of the below isosceles triangle:

*h² *– 4² = 5.8²

*h²* = 5.8² – 4²

*h²* = 33.64 – 16

*h²* = 17.64

*h* = √17.64

*h* = 4.2 cm

Point to note:

- An the height of an isosceles triangle divides it into two equal parts, therefore the length of the right angle section must be half of the base.

**Finding the height of an equilateral triangle**

The Pythagoras Theorem can also be used to calculate the height of an equilateral triangle. For example:

*h² +* 2² = 4²

*h²* = 4² – 2²

*h²* = 16 – 4

*h²* = 12

*h* = √12

*h* = 3.46 cm (to 2 d.p.)

**Applying Pythagoras’ Theorem twice**

There maybe scenarios where the Pythagoras Theroem is required to be applied twice to find a required length. For example, in the example below, the length of *b* is required to be calculated but the length of *a* can be calculated.

Find the length of *b* first

*b² =* 18² – (4 + 9)²

*b² =* 324 – 169

*b² =* 155

*a² =* *b*² + 4²

*a² =* 155 + 4²

*a² =* 171

*a =* 13.08 cm (to 2.d.p.)

**Finding the distance between two points**

Pythagoras’ Theorem can be used to find the distance between any two points on a co-ordinate grid.

For example, find the distance between the points A(7, 6) and B(3, -2). (AC is the vertical distance and BC is the horizontal distance).

AB*² =* AC² + BC²

AB*² =* 8² + 4²

AB*² =* 64 + 16

AB*² =* 80

AB*² =* 8.94 units (to 2 d.p.)