# Right Angled Triangle

## Definition

A triangle whose one of three angles is right angle, is called a right angled triangle.

Right angled triangle is a triangle and it is also formed by the three line segments but two of three line segments meet each other perpendicularly. Therefore, the angle between them is a right angle ($90^\circ$). Hence, the triangle is called as a right angled triangle.

### Sides

The three sides of the right angled triangle are called by three special names but there is no significance for the sides of the other triangles.

1

#### Opposite side

The side which displayed in vertical direction is called opposite side. It is also called as perpendicular because it is perpendicular to the surface of the earth.

For example, $\overline{BC}$ is the opposite side of the right angled triangle $BAC$.

2

The side which displayed in horizontal direction is called adjacent side. It is also called as base because it is parallel to the surface of the earth.

For example, $\overline{AB}$ is the adjacent side of the right angled triangle $BAC$.

3

#### Hypotenuse

The side which displayed in slant and opposite to the right angle is called hypotenuse.

For example, $\overline{AC}$ is the hypotenuse of the right angled triangle $BAC$.

### Angles

Three interior angles are formed in right angled triangle by the meeting of every two of three sides.

1

#### Right angle

One of three interior angles is a right angle and it is the angle, formed by the meeting of opposite and adjacent sides.

For example, $\angle CBA$ (or) $\angle ABC$ (or) $\angle B = 90^\circ$

2

#### Angle of the Right angled triangle

It is an interior angle which is an acute angle but it is the angle between hypotenuse and adjacent side. It is called as angle of the right angled triangle.

For example, $\angle BAC$ (or) $\angle CAB$ (or) $\angle A = \theta$.

3

#### Other angle

It is also an interior angle and acute angle but it is the angle between opposite side and hypotenuse. There is no much importance to this angle.

For example, $\angle BCA$ (or) $\angle ACB$ (or) $\angle C$

##### Construction

A right angled triangle can be constructed geometrically by using geometric tools.

1. Take ruler and draw a horizontal line of any length. In this example, $11$ centimetres line is drawn horizontally. The left and right side endpoints of line segments are called as point $D$ and $E$.
2. Take protractor. Coincide centre of the protractor with point $D$ and then coincide right side base line with horizontal line $\overline{DE}$. After that identify $90^\circ$ angle by considering bottom scale and mark it.
3. Take ruler and draw a straight line from point $D$ through $90^\circ$ angle line.
4. Take compass and adjust it to have a particular distance (In this example $8$ Centimetres) between points of pencil lead and needle by considering the ruler. Draw an arc from point $D$ on $90^\circ$ angle line. The arc cuts the $90^\circ$ angle line at a particular point and call it as point $F$.
5. Take ruler and join points $E$ and $F$.

The step by step geometrical procedure constructed a triangle, in which the points $D$, $E$ and $F$ are called as vertices. The line segments $\overline{DE}$, $\overline{EF}$ and $\overline{FD}$ are called sides.

1
###### Lengths of sides

The three sides are called by different names.

1. The side $\overline{DF}$ is called opposite side and its length is $DF = 8$ Centimetres.
2. The side $\overline{DE}$ is called adjacent side and its length is $DE = 11$ Centimetres.
3. The side $\overline{EF}$ is called hypotenuse but its length ($EF$) is unknown. Take ruler and measure it. It will be $EF = 13.6$ centimeters exactly.
2
###### Angles

The triangle has three interior angles.

1. The angle $\angle EDF$ (or) $\angle FDE$ (or) $\angle D$ is a right angle. It can be expressed as $\angle D = 90^\circ$.
2. The angle $\angle DEF$ (or) $\angle FED$ (or) $\angle E$ is the angle of the right angled triangle but it is unknown. Measure the angle by using protractor. It will be $\angle E = 36^\circ$.
3. The third interior angle $\angle DFE$ (or) $\angle EFD$ (or) $\angle F$ is unknown. So, measure it by using protractor and it will be $\angle F = 54^\circ$ exactly.