Right-Angled Triangle Properties

We are given a right-angled triangle with the right angle at vertex A ($\angle A = 90^\circ$). We are also given that angle B is $45^\circ$ ($\angle B = 45^\circ$).

The sum of angles in a triangle is $180^\circ$. Therefore, angle C can be calculated as:

$\angle C = 180^\circ - \angle A - \angle B = 180^\circ - 90^\circ - 45^\circ = 45^\circ$.

Since $\angle B = \angle C = 45^\circ$, the triangle is an isosceles right-angled triangle. In an isosceles triangle, the sides opposite the equal angles are equal in length. Thus, the side opposite $\angle C$ (which is side AB) is equal to the side opposite $\angle B$ (which is side AC).

• $AB = AC$

Given $AB = 3$ m, it follows that $AC = 3$ m.

Calculating Hypotenuse BC using Pythagoras

The side BC is the hypotenuse of the right-angled triangle.

According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (legs).

The theorem is expressed as:

$AB^2 + AC^2 = BC^2$

Substitute the known lengths of AB and AC:

$ (3 \text{ m})^2 + (3 \text{ m})^2 = BC^2 $

$ 9 \text{ m}^2 + 9 \text{ m}^2 = BC^2 $

$ 18 \text{ m}^2 = BC^2 $

To find the length of BC, take the square root of both sides:

$ BC = \sqrt{18 \text{ m}^2} $

Simplify the square root:

$ BC = \sqrt{9 \times 2} \text{ m} = 3\sqrt{2} \text{ m} $

Alternative Calculation using Trigonometry

We can also use trigonometry. We know $\angle B = 45^\circ$, the adjacent side $AB = 3$ m, and we want to find the hypotenuse $BC$. The cosine function relates these:

$ \cos(\angle B) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{AB}{BC} $

Substitute the known values:

$ \cos(45^\circ) = \frac{3 \text{ m}}{BC} $

Since $\cos(45^\circ) = \frac{1}{\sqrt{2}}$:

$ \frac{1}{\sqrt{2}} = \frac{3 \text{ m}}{BC} $

Solving for $BC$:

$ BC = 3\sqrt{2} \text{ m} $

Both methods confirm the length of side BC.