There are two fundamental relations between sides of a right triangle when its angle equals to $60$ degrees.

- The length of adjacent side is equal to half of the length of hypotenuse.
- The length of opposite side is equal to $\small \sqrt{3}/{2}$ times of the length of hypotenuse.
- The third angle of right triangle is $\small 30^°$.

Draw a horizontal straight line $\overrightarrow{ST}$, and then draw a $60$ degrees line from point $S$ in anticlockwise direction. Set compass to any length with the help of a ruler and then draw an arc on $60$ degree line from $S$ and their intersecting point is $U$. After that, use protractor and draw a perpendicular line to straight line $\overrightarrow{ST}$ from point $U$ and take, it intersects the straight line $\overrightarrow{ST}$ at point $V$. Thus a right angled triangle $VSU$ is formed with an angle of $60^\circ$.

Take, the length of the side $\overrightarrow{SU}$ is $d$. but the length of the adjacent side $\overline{SV}$ will be $\dfrac{d}{2}$. Now, evaluate the length of the opposite side $\overline{UV}$ and it can be done by using Pythagorean Theorem.

${SU}^2 \,=\, {SV}^2 + {UV}^2$

$\implies UV \,=\, \sqrt{{SU}^2 -{SV}^2}$

$\implies UV \,=\, \sqrt{d^2 -{\Bigg(\dfrac{d}{2}\Bigg)}^2}$

$\implies UV \,=\, \sqrt{d^2 -\dfrac{d^2}{4}}$

$\implies UV \,=\, \sqrt{\dfrac{4d^2 -d^2}{4}}$

$\implies UV \,=\, \sqrt{\dfrac{3d^2}{4}}$

$\implies UV \,=\, \dfrac{\sqrt{3}d}{2}$

$\,\,\, \therefore \,\,\,\,\,\, UV \,=\, \dfrac{\sqrt{3}}{2} \times {SU}$

The two properties of the sides of right angled triangle can be proved practically when the angle of right angled triangle is $60^\circ$.

Draw a horizontal ray $\overrightarrow{AB}$ and then draw a $60$ degrees line in anticlockwise direction from point $A$. After that, set compass to any length with the help of a ruler. In this case, compass is set to $10 \, cm$ and draw an arc on $60^\circ$ line from point $A$ and their intersecting point is $C$.

Finally, draw a perpendicular line to ray $\overrightarrow{AB}$ from point $C$ and it intersects the ray $\overrightarrow{AB}$ at point $D$. Thus, a right angled triangle, called $\Delta DAC$ is formed geometrically.

Take ruler and check the length of adjacent side ($\overline{AD}$) and you observe that it will be $5 \, cm$ exactly. It is proved that the length of adjacent side is exactly half of the length of the hypotenuse of right angled triangle when the angle of triangle is $60^\circ$.

Now, take ruler and measure the length of the opposite side $\overline{CD}$. You will observe that it will be more than $8.65 \, cm$ but very close to $8.7 \, cm$.

Theoretically, the value of opposite side is $\dfrac{\sqrt{3}}{2} \times 10 \,=\, 8.6602$

Hence, the length of the opposite side is more than $8.65 \, cm$ but less than $8.7 \, cm$.

Therefore, it is proved that the length of opposite is $\dfrac{\sqrt{3}}{2}$ times to length of hypotenuse when the angle is equal to $60^\circ$.

Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers.
Know more

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.