# Right Triangle

Triangle whose one angle is exactly 90 degree is known as Right Triangle

Why its called Right Triangle?
The name is derived from right angle whose measure is 90 degree.

Examples of Right angle triangle
Given below are examples of right triangle.
Note all of them contain angle measuring exactly 90 degree

## Structure of Right Triangle

Right triangle contain following components:

(A) Hypotenuse
The side opposite to 90 degree angle is called Hypotenuse
Its also the longest side of the triangle

(B) Base
The bottom side of the triangle is known as base

(C) Height
The line perpendicular to the base is called height of the right triangle.
Note that base and height are perpendicular (i.e. 90 degree) to each other

While naming the right triangle, you have to crucial to identify Hypotenuse correctly, the other arms “base” and “height” can be used interchangeably

Let us observe some examples relate to Right Triangle structure

Example 01

Above image is of right triangle ABC, where;
Side BC = Base
Side AB = Height
Side AC = Hypotenuse

Observe that:
Height and Base are perpendicular to each other
Hypotenuse is the longest side

Example 02

The above image of right triangle is tricky because its tilted.

Here the side BC is at the bottom but it cannot be called Base because it is opposite to the 90 degree angle and also the longest side of the triangle

Hence,
Side BC = Hypotenuse
The other sides are base and height, both can be named interchangeably

## Angles of Right Triangle

In right triangle, one angle measures exactly 90 degree while the other two angles are acute angle.

In all the above examples, note one of the angle is measures 90 degree while other angles in blue color are acute angle.

Figure (A) : Angle B is 90 degree
Figure (B) : Angle A measures 90 degree
Figure (C) : Angle C measures 90 degree

## Area of Right Triangle

The formula for right triangle area calculation is very easy and straight forward

Area=\ \frac{1}{2} \times \ base\ \times height

## Property of Right Triangle

(01) Right triangle contain one 90 degree angle and two acute angle

(02) Pythagoras Theorem
It says that square of hypotenuse is equal to sum of square of other two sides

Consider the below right triangle ABC

Pythagoras Theorem states that:
( hypotenuse)^{2} =\ \ height^{2} +base^{2}

( hy)^{2} =\ \ h^{2} +b^{2}

This is one of the most important concepts of right angle triangle. Please practice the formula on paper in order to remember it for your exams.

Let us solve some problems related to the concept:

Example 01
Below right angle triangle ABC, Base = 3 cm and Height = 4 cm.
Find the length of hypotenuse

( hy)^{2} =\ \ h^{2} +b^{2}\\\ \\ ( hy)^{2} =\ \ 4^{2} +3^{2}\\\ \\ ( hy)^{2} =\ \ 16\ +\ 9\\\ \\ ( hy)^{2} =\ 25\\\ \\ hy\ =\ 5

Hence, length of Hypotenuse is 5 cm

Example 02
Given below is right angled triangle with height 5 cm and hypotenuse 8 cm.
Find the measurement of the third side

Using Pythagoras Theorem

(hy)^{2} =\ \ h^{2} +b^{2}\\\ \\ ( 8)^{2} =\ \ 5^{2} +b^{2}\\\ \\ 64-25=\ \ b^{2}\\\ \\ ( b)^{2} =\ 39\\\ \\ b\ =\ 6.24

(03) Circumcenter of Right Triangle
The circumcenter of right triangle lies at the midpoint of the hypotenuse.

What is Circumcenter?
It is the center of circle which passes through all the three vertices of triangle.
The circumcenter can be located by the intersection of perpendicular bisector of triangle sides

Locating Circumcenter of Right Triangle
Below is the right triangle of which we locate the circumcenter

Step 01
Draw perpendicular bisector of side BC

Step 02
Similarly draw perpendicular bisector of side BA

The intersection point of the perpendicular bisector (point O) is the circumcenter of the triangle.
Note the point O is just at the middle of hypotenuse.

Below is how circumcircle looks like for the triangle

The radius of circumcircle is half the hypotenuse

Conclusion
The circumcenter of right triangle lies at the midpoint of hypotenuse of the triangle

(04) Centroid of the Right Triangle
Centroid of right triangle lie inside the triangle

What is centroid?
Centroid is the middle point of any graphical figure.
In triangle centroid is located at the intersection of medians.

Note: Centroid divides median in the ration 2:1

Locating Centroid of Right Triangle
Step 01
Find the midpoint of side BC and draw a line touching midpoint and opposite vertex of triangle
This line MA is the median of side BC

Step 02
Similarly draw the median from side AB

The point of intersection of two median (i.e point O) is the centroid of the triangle ABC

Median & Centroid Point
Note that the centroid divides the median in the ratio of 2 : 1

Suppose the length of median CN be x cm
Then length of CO = 2/3 x
Length of ON = 1/3 x

Similar is the case for other median lines.

Conclusion
The median of right triangle is inside the triangle

## Frequently asked Question – Right Triangle

(01) Is the below image is of Right triangle?

No, It is not a right triangle.
For Right Triangle the angle measurement has to be exactly 90 degrees

(02) Can Right Obtuse angle exists?

No !!
If one angle is 90 degrees, the other two angles should be acute angles, otherwise the following rule of triangle will not satisfy.
Sum of three angle of triangle = 180 degree

(03) Can a triangle have two 90 degree angle?

Not Possible.

We know that:
sum of interior angle = 180

When there are two 90 degree angle, the above rule will not be satisfied as there will be no room left for third angle.

⟹ 90 + 90 + x = 180
⟹ 180 + x = 180
No room for angle x !!

(04) If length of two sides of right triangle is given. How to find the length of third side?