Math Doubts

Properties of Right angled triangle when its angle is 0°


Right angled triangle has two special properties when angle of the right angled triangle is $0^\circ$.

  1. The length of the opposite side is zero
  2. The length of the adjacent side is exactly equal to the length of the hypotenuse.


The properties of the right angled triangle can be proved mathematically from two geometrical approaches.


Fundamental approach

Every right angled triangle has an angle and also has an angle of $0^\circ$ in a special case. The geometric shape of a right angled triangle should be known in order to know the properties of the right angled triangle when the angle of the right angled triangle is $0^\circ$.

$\Delta CAB$ is an example triangle for the right angled triangle of an angle $0^\circ$. It is initially having an angle $\theta$ but later its angle is reduced to zero by reducing the length of the opposite side. When the angle of the right angled triangle is adjusted to zero degrees, the structure of the right angled triangle is transformed as a line segment. At the angle of $0^\circ$, the length of the opposite side will be zero because the both end points of opposite side lie on same position in the plane.

$Length \, of \, Opposite \, side = 0$

When the angle of the right angled triangle is zero, there is no angle between adjacent side and hypotenuse. Moreover, the lengths of both adjacent side and hypotenuse are equal geometrically.

$Length \, of \, Adjacent \, side$ $=$ $Length \, of \, Hypotenuse$


Practical approach

Construct a right angled triangle with its angle is zero degrees by using geometric tools.

8 centimeters right angled triangle with 0 degrees angle
  1. Take ruler and draw a horizontal line of any length and call its left side point as Point $D$.
  2. Use protractor, coincide its centre with point $D$ and also coincide its right side base line with horizontal line. Now, identify zero degrees by considering bottom scale. Surprisingly, it also lies on same horizontal line because the horizontal line is a representation of zero angle geometrically. However, just mark the $0^\circ$ angle.
  3. Take ruler and draw a line from point $D$ through $0^\circ$ angle marked point but the line is in the same direction of the horizontal line.
  4. Take compass and set distance between points of needle to pencil lead to any length. Here, the length is $8$ centimetres as an example. Then, draw an arc from Point $D$ on $0^\circ$ angle line and it cuts the line at point $E$.
  5. A perpendicular line should be drawn from point $E$ to horizontal line to get the right angled triangle but it is not possible to do it because the $0^\circ$ angle line and horizontal line are in same positions in the plane but imagine you have drawn a perpendicular line from $E$ to horizontal line and assume it meets the horizontal line at point $F$ to obtain the structure of the right angled triangle.

Thus, the right angled triangle ($\Delta FDE$) is constructed but the points $E$ and $F$ are at same position on the plane. The line segments $\overline{DE}$, $\overline{EF}$ and $\overline{DF}$ become hypotenuse, opposite side and adjacent side respectively.

Observe the right angled triangle $FDE$ and its angle is $0^\circ$. The length of the opposite side ($EF$) is zero but the lengths of hypotenuse ($DE$) and adjacent side ($DF$) are equal.

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