Right triangle is the most important concept in geometry and has many applications from decades. A right triangle is a triangle having a right angle (90$^{0}$) in it.

Trigonometry serves as a basis to find the relation between the sides and angles of a right triangle. Greek mathematician named Pythagoras developed a formula called the Pythagorean theorem for finding the lengths of the sides of any triangle.  

1. The formula to find the area of a right triangle is

Area = $\frac{1}{2}$ (b $\times$ h)

where b : base of the triangle and h : height of the triangle

2. Using Pythagoras formula we can find the unknown side in a right angled triangle.
c$^{2}$ = a$^{2}$ + b$^{2}$
where 'c' is the long side in a triangle and 'a', 'b' are the legs of the triangle.

3. The unknown angles of a right triangle can be found by the formula

A = $\sin^{-1}$ ($\frac{a}{c}$) = $\cos^{-1}$ ($\frac{b}{c}$) = $\tan^{-1}$ ($\frac{a}{b}$)

where, a, b and c are the sides of a triangle.
1. Acute angles of a right triangle are complementary.
  • A + C + 90$^{0}$ = 180$^{0}$
  • $\Rightarrow$ A + C = 90$^{0}$
  • $\Rightarrow$ A = 90$^{0}$ - C.
Right Angle Triangle

2. The hypotenuse is always across from the right angle.
3. The hypotenuse will be twice the length of the shorter side if the acute angle is measuring 30 and 60 degrees.
4. In geometry according to Side- Angle-Side rule: If two sides and the included angle of a triangle are congruent to two sides and the included angle of another triangle. Then the two triangles are said to be congruent.
Special right triangle is a triangle with some features that makes calculations easier and will yield exact answers. There are two types of special right triangles 45$^{0}$ - 45$^{0}$- 90$^{0}$ and 30$^{0}$ - 60$^{0}$ - 90$^{0}$.

1. 30$^{0}$ - 60$^{0}$ - 90$^{0}$: This right triangle has unique ratio of its sides and the ratio of the sides of a right angle triangle is 1: $\sqrt{3}$ : 2

Here the angles will be in arithmetic progression.
Triangles
where H : Hypotenuse
LL : Long leg (Across from 60$^{0}$)
SL : Short leg (Across from 30$^{0}$)
Short Triangle
The formula for short leg is
SL = $\frac{1}{2}$ H

The formula for long leg is
LL = $\frac{1}{2}$ H $\sqrt{3}$

Now combining the first two we get

LL = SL $\sqrt{3}$

2. 45$^{0}$ - 45$^{0}$- 90$^{0}$: In this triangle the angles will be in the ratio 1 : 1 : 2 and the sides are in the ratio 1 : 1 : $\sqrt{2}$.

Isosceles Right Triangle
Isosceles Triangles
Two triangles will be said similar if the only difference is size. Similar right angled triangles will have their angles equal and sides will have the same common ratio. Similar triangles are also called equiangular triangles as they will have same shape but may differs in size.

As the sum of the angles in a triangle is 180$^{0}$, the sum of two acute angles in a right triangle is 90$^{0}$.
Similar Right Triangle

In this figure, $\triangle$ABC $\sim$ $\triangle$ A'B'C'

If the angles A and A' are equal then B and B' will also be equal.
A triangle has three sides and three angles. The angles add up to 180$^{0}$ and each angle of a equilateral triangle is of 60$^{0}$.

As a right triangle should have a right angle in it.

However, we cannot have an equilateral right angled triangle.
In an Isosceles right triangle two sides and two angles will be equal. The angles are $45^{\circ}-45^{\circ}-90^{\circ}$ and the isosceles right triangle satisfies the pythagorean theorem. The sides of an isosceles right triangle will be in the ratio of 1: 1: $\sqrt{2}$. The longest side of a right triangle is the hypotenuse and the other two sides are known as legs.
Triangle

Where H : Hypotenuse and L : Leg
Right Angle Triangle
The formulas for H and L is given by:

H = L $\sqrt{2}$

L = $\frac{1}{2}$ (H $\sqrt{2}$)

In a scalene triangle all the sides will be of different length. In a right scalene triangle one of the angles will be 90$^{\circ}$ and the sides will be of different lengths.

A scalene triangle will have no congruent sides and has no line of symmetry. The angles in a scalene right triangle will not be equal.
In general, centroid is a point where all the three medians of the triangle intersect.
Consider the right triangle given above having the hypotenuse y = ax

Right Angled Triangle Centroid
As we know the formula for area of a triangle which is Area = $\frac{1}{2}$ (b$\times$ h)

$\Rightarrow$ Area= $\frac{1}{2}$ (a $\times$ b$^{2}$)

From the figure we can write it as

X $\times$ area = $\int_{0}^{b}x\ da$

where da = ydx

So now,
X $\times$ area = $\int_{0}^{b}x\ da$

X $\times$ area = a $\int_{0}^{b}$ x$^{2}$ dx

X $\times$ area = $\frac{ab^{3}}{3}$

Now X = $\frac{\frac{ab^{3}}{3}}{area}$

X = $\frac{2b}{3}$

Therefore centroid of a right triangle is X = $\frac{2b}{3}$.
Following illustrates the use of right triangles in real life
  1. Construction of a staircase involves right triangles.
  2. Real life example of equilateral triangle is traffic signal.
  3. Peak of the roof often forms an obtuse triangle.
  4. Paper footballs are right triangles.
  5. Stationary objects do form right triangle which can be understood by using right triangle solving techniques. In real life we can see it when a shadow is cast by a tall object. It is immensely helpful to draw a picture of the situation to solve the problem easily.

Solved Examples

Question 1: If the longer leg of a 30 - 60 - 90 right triangle is 7 $\sqrt{3}$ then find the length of the hypotenuse.

Hypotenuse
Solution:
 
x (shorter leg) is opposite to 30$^{\circ}$ and x $\sqrt{3}$ the longer leg is opposite to 60$^{\circ}$. As x $\sqrt{3}$ = 7 $\sqrt{3}$ the hypotenuse is 2x and x = 7.

$\Rightarrow$ 2x =14.

Therefore the hypotenuse is 14.
 

Question 2: Which of the following is an isosceles right triangle?
Isosceles Triangles


a) Figure 1
b) Figure 3
c) Figure 4
d) Figure 2
Solution:
 
In an isosceles right triangle two sides will be equal and one of the angle will be 90$^{\circ}$.
So the answer is 'd' figure 2.