Right angled triangle has two special properties when the angle of the triangle is $45^\circ$.

- The length of the opposite side is equal to the length of adjacent side.
- The length of the hypotenuse is equal to $\sqrt{2}$ times of the length of the opposite side or adjacent side.

The two properties of the right angled triangle when the angle of the triangle is $45^\circ$, can be proved mathematically in two different approaches.

1

Construct a right angled triangle by drawing a line of any length with $45^\circ$ angle and observe the lengths of the opposite side and adjacent side. Also observe the ratio of length of the opposite side (or adjacent side) to length of the hypotenuse.

As an example, a right angled triangle is constructed by drawing a line of $4$ centimeters with $45^\circ$ degrees angle. Just follow below steps to construct it your own with geometric tools.

- Draw a line in horizontal direction on the paper. Call left side point of the horizontal line as point $P$.
- Take protractor.
- Coincide point $P$ with middle point of the protractor.
- Coincide the horizontal line with right side base line of the protractor.
- Consider bottom angle scale of the protractor and mark a point at angle $45^\circ$.
- Draw a line from point $P$ through marked point at angle $45^\circ$ by using scale.
- Take compass.
- Set length of the compass to $4$ centimetres with the help of scale.
- Draw an arc from point $P$ on the $45^\circ$ angle line. The arc cuts the $45^\circ$ line at a point and assume to call the point as point $Q$.
- Take set square and draw a line from point $Q$ but it should be perpendicular to the horizontal line. The line, drawn from point $Q$ intersects the horizontal line perpendicularly at a point and it is called as point $R$.

Thus a right angled triangle $(\Delta QPR)$ is constructed geometrically. Now, take scale and measure length of the opposite side and also length of the adjacent side. It is observed that the length of the opposite side is equal to the length of the adjacent side and the length of the side is measured as $2.85$ centimetres.

Calculate the ratio of length of the opposite side (or adjacent side) to length of the hypotenuse.

$$\frac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse} = \frac{2.85}{4} = 0.7125$$

Repeat same procedure and construct a right angled triangle by drawing a line of $5$ centimeters with an angle $45^\circ$. You get that the length of the opposite side and length of the adjacent side are equal and the value of them is $3.55$ centimetres roughly.

Calculate the ratio of the length of the opposite (or adjacent side) to length of the hypotenuse.

$$\frac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse} = \frac{3.55}{5} = 0.71$$

Repeat the procedure one last time to construct a right angled triangle by drawing a line of $9$ centimeters with angle $45^\circ$. You observe that the length of the opposite side is equal to length of the adjacent side, and the length is $6.35$ centimeters.

Calculate the ratio of the length of the opposite side (or adjacent side) to length of the hypotenuse.

$$\frac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse} = \frac{6.35}{9} = 0.7055$$

The three values, which obtained from the ratio of length of the opposite side (or adjacent side) to length of the hypotenuse in each case, are approximately same when the angle of the right angled triangle is $45^\circ$. It is possible if the length of the opposite side (or adjacent side) is directly proportional to the length of the hypotenuse.

You can repeat the procedure as many times as you want to construct right angled triangle by drawing a line of any length with angle $45^\circ$ and you get similar values but none of them is absolute. However, it can be obtained exactly by the algebraic approach with geometric considerations.

2

In $\Delta QPR$, it is assumed that

- The length of the opposite side $(QR)$ is $h$.
- The length of the adjacent side $(PR)$ is $l$.
- The length of the hypotenuse $(PQ)$ is $r$.

According to Pythagorean Theorem

$r^2 = l^2 + h^2$

Geometrically, it is proved that the length of the opposite side is equal to length of the adjacent side if angle of the right angled triangle is $45^\circ$. Therefore, $l = h$.

$\implies r^2 = l^2 + l^2$

$\implies r^2 = 2l^2$

$\therefore \,\, r = \sqrt{2}.l$ (or) $r = \sqrt{2}.h$

It is proved that the length of the hypotenuse is proportional to length of the opposite side or adjacent side in the case of right angled triangle whose angle is $45^\circ$, and it is $\sqrt{2}$ times to length of both opposite side and adjacent side.

According to this, the ratio of length of hypotenuse to length of opposite side or adjacent side is,

$$\frac{r}{l} = \frac{1}{\sqrt{2}}$$

$$\implies \frac{r}{l} = 0.707106781…$$

Geometrically, we got $0.7125$, $0.71$ and $0.7055$ as the approximate values of ratio of length of opposite side to length of hypotenuse in three cases if angle of right angled triangle is $45^\circ$ but the exact value is $\frac{1}{\sqrt{2}}$ or $0.707106781…$

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

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