Right angled triangle has one special property when the angle of the right angled triangle is equal to $30^\circ$. The length of the opposite side is equal to half of the length of the hypotenuse in every right angled triangle whose angle is $30$ degrees. Similarly, the length of the hypotenuse is twice the length of the opposite side.

It can be proved geometrically by constructing a right angled triangle of $30$ degrees angle with any lengths of sides. Let us verify this property by constructing three different right angled triangles with $30^\circ$ angle and different lengths using geometric tools.

1

Just follow below steps to construct a right angled triangle with $30^\circ$ angle and $6$ centimetres length.

- Take Ruler and draw a horizontal line. Call left side point of the horizontal line as point $A$.
- Take protractor. Coincide the point $A$ with centre of the protractor and also coincide the horizontal line with right side base line of the protractor. Identify $30^\circ$ angle by considering bottom same scale of the protractor and mark it.
- Take ruler again and draw a straight line from point $A$ through the $30^\circ$ angle marked point.
- Take compass and adjust it to get $6$ centimetres length between needle point and pencil lead with the help of ruler. Draw an arc from point $A$ on $30^\circ$ angle line and it cuts the $30^\circ$ angle line at point $B$.
- Take set square and draw a perpendicular straight line from point $B$ to horizontal line and the perpendicular line intersects the horizontal line perpendicularly at point $C$.

The right angled triangle $CAB$ is constructed as the result of this geometric procedure.

In $\Delta CAB$, the length of the hypotenuse is $6$ centimetres but the length of the opposite side is unknown. So, take ruler and measure the length from point $B$ to $C$. It will be $3$ centimeters exactly.

It means, the length of the opposite side is half of the length of the hypotenuse. In other words, the length of the hypotenuse is twice the length of the opposite side.

2

Use same geometrical procedure and construct a right angled triangle ($\Delta EDF$) with $30^\circ$ angle and $7$ centimetres length line.

In $\Delta EDF$, the length of the hypotenuse is $7$ centimetres. Measure the length of the opposite side by using ruler. It will be $3.5$ centimetres exactly.

In the case of $\Delta EDF$ also, the length of the opposite side is equal to half of the length of the hypotenuse, and the length of the hypotenuse is twice the length of the opposite side.

3

Construct right angled triangle $\Delta IGH$ with $30^\circ$ angle and $10$ centimetres line by using same geometrical process.

The length of hypotenuse is $10$ centimetre and measure the length of opposite side by using ruler. Surprisingly, it will be $5$ centimetres exactly.

It is also proved in $\Delta IGH$ that length of the opposite side is exactly equal to half of the length of the hypotenuse, and the length of the hypotenuse is equal to twice the length of opposite side.

In these three example cases, it is proved that the length of the hypotenuse is equal to twice the length of the opposite side and the length of the opposite side is equal to half of the length of the hypotenuse. It is possible in right angled triangles ($\Delta CAB$, $\Delta EDF$ and $\Delta IGH$) because all three right angled triangles have $30^\circ$ as angle commonly.

You can also test this property by constructing right angled triangle with $30^\circ$ angle.

If angle of the right angled triangle is $30$ degrees,

- $$Length \, of \, Hypotenuse = 2 \times Length \, of \, Opposite \, side$$
- $$Length \, of \, Opposite \, side = \frac{Length \, of \, Hypotenuse}{2}$$

On the basis of the relation between opposite side and hypotenuse, the relation of hypotenuse with adjacent side can be derived by using Pythagoras theorem.

$\Delta POQ$ is a right angled triangle and its angle is $30^\circ$. It is proved that the length of the opposite side is half of the length of the hypotenuse in this type of triangles. Assume, length of the hypotenuse is $d$. So, the length of opposite side should be $\frac{d}{2}$.

Apply Pythagorean Theorem to this triangle for obtaining the length of the adjacent side in terms of length of hypotenuse.

$$d^2 = {\Bigg(\frac{d}{2}\Bigg)}^2 + OQ^2$$

$$\implies d^2 \,-\, {\Bigg(\frac{d}{2}\Bigg)}^2 = OQ^2$$

$$\implies OQ^2 = d^2 \,-\, {\Bigg(\frac{d}{2}\Bigg)}^2$$

$$\implies OQ^2 = d^2 \,-\, \frac{d^2}{4}$$

$$\implies OQ^2 = d^2 \Bigg[1 \,-\, \frac{1}{4}\Bigg]$$

$$\implies OQ^2 = d^2 \Bigg[\frac{3}{4}\Bigg]$$

$$\implies OQ^2 = \Bigg[\frac{3}{4}\Bigg]d^2$$

$$\implies OQ = \sqrt{\Bigg[\frac{3}{4}\Bigg]d^2}$$

$$\implies OQ = \frac{\sqrt{3}}{2}d$$

$$\implies OQ = \frac{\sqrt{3}}{2} \times d$$

$$\implies OQ = \frac{\sqrt{3}}{2} \times OP$$

$$\therefore \,\, Length \, of \, Adjacent \, side = \frac{\sqrt{3}}{2} \times Length \, of \, Hypotenuse$$

It is proved that the length of the adjacent side is equal to $\frac{\sqrt{3}}{2}$ times the length of the hypotenuse in the case of the right angled triangle whose angle is $30^\circ$.

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