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Right Triangle

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A right triangle or right-angled triangle is a triangle which contains a right angle, whereas the right angle is an angle measuring $90^{\circ}$. The following figure shows a right-angled triangle:
Right Triangle
In the above figure, $\angle PQR$ is a right angle. Side PR opposite to right angle is known as hypotenuse. The side PQ is called perpendicular or height and other side QR is called base.
Hypotenuse is the longest side in a right triangle.
The sides and angles of right triangles are related in many ways. Various formulas and theories are derived on the basis of right triangle.

Right Triangle Definition

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The definition of a right triangle is as follows:
A triangle is known as right triangle, if it has one angle as right angle.
Right Triangle Definition
The right angle in a right triangle is shown either by notation $90^{\circ}$ or by the symbol as shown in the figure above (mark in red color).

Right Triangle Formula

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In a right triangle, $\angle C=90^{\circ}$. Let us assume that side opposite to $\angle C$ is "c", side opposite to $\angle A$ is "a" and side opposite to $\angle B$ is denoted by "b".
Right Triangle Formula
A theorem known as Pythagoras Theorem is defined in right-angled triangle. It states that, "In a right triangle, square of the longest side or the hypotenuse is equal to the sum of squares of other two sides."

Pythagoras Theorem is expressed as:
Right Triangle Formulas
Area of a right triangle is given by:
Area of Right Triangle Formula

Perimeter of right triangle is given by:
Preimeter of Right Triangle Formula

Right Triangle Rules

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Many rules are defined on the basis of right triangle. In fact, one whole branch of mathematics is based on right triangles. That branch is known as trigonometry which is all about right triangles and their properties.

Trigonometric ratios are defined in right triangle as shown below:
Let us consider a right triangle $\bigtriangleup ACB$ right angled at C. If $\angle B$ is named as $\theta $.
Right Triangle Rules
Then, following rules hold:
$\sin \theta$ = $\frac{AC}{AB}$

$\cos \theta$ = $\frac{BC}{AB}$

$\tan \theta$ = $\frac{AC}{BC}$

$\csc \theta$ = $\frac{AB}{AC}$

$\sec \theta$ = $\frac{AB}{BC}$

$\cot \theta$ = $\frac{BC}{AC}$

Sine Rule:

Right Triangle Sine Rule
Cosine Rule:
Right Triangle Cosine Rule

Equilateral Right Triangle

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A right triangle can never be an equilateral triangle, because according to the definition of triangle, sum of all three angles must be $180^{\circ}$. While, by the definition of equilateral triangle, it has equal sides and equal angles. So logically, if we try to construct an equilateral right triangle, then it must be having all of its angles as $90^{\circ}$. But, when we find the sum of three angles of such imaginary triangle, we get
$90^{\circ}+90^{\circ}+90^{\circ}=270^{\circ}$
which is greater than $180^{\circ}$. Hence, an equilateral right triangle is not at all possible.

Acute Right Triangle

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An acute triangle is one which has all its angles measuring less than $90^{\circ}$. But, in a right triangle, one angle must be $90^{\circ}$. So, an acute right triangle will not be practically possible, since a right triangle must be having one right angle and two acute angles.

Right Obtuse Triangle

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An obtuse triangle is one which has at least one of its angles measuring greater than $90^{\circ}$. But, in a right triangle, one angle must be $90^{\circ}$. So, an obtuse triangle will also not be practically possible, since a right triangle must be having one right angle and two acute angles.

If we try to hypothetically construct a right triangle with an obtuse angle, then sum of all three angles will be greater than $180^{\circ}$. Hence, a right obtuse triangle is not possible.

Right Triangle in Real Life

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Right triangles are everywhere in real life. They are all around us. There are many such examples. Few are as follows:
  • If we place a ladder against the wall, it makes a right triangle with the wall.

Right Triangle in Real Life

  • While finding distance or angle of elevation from a ship to a tall tower, it forms right angle.

Right Triangles in Real Life

  • A sandwich is in the shape of right triangle.

Right Triangles

  • A triangle made by staircase and wall is always a right triangle.

Right Triangle in Real World
There are many many more examples of right triangles which we come across in real life.

Right Triangle Word Problems

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Given below are few word problems of right triangles.

Solved Examples

Question 1: The angle of elevation of a 10-meter high building from a fixed point is noted to be $60^{\circ}$. Find the distance of the point from the base of the building.
Solution:
Step 1: The following figure is obtained according to the question:
Right Triangle Word Problems

Let us assume that the distance of given point from the base of the building is x meter.
According to a property of right triangle, we have
$\tan \theta$ = $\frac{AB}{BC}$

$\tan 60^{\circ}$ = $\frac{10}{x}$

$\sqrt{3}$ = $\frac{10}{x}$

$x$ = $\frac{10}{\sqrt{3}}$

$x$ = $\frac{10}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}$

= $\frac{17.32}{3}$

= 5.8 meter (approx)

Step 2: The following figure is obtained according to the question:
Right Triangle Word Problems

Let us assume that the distance of given point from the base of the building is x meter.
According to a property of right triangle, we have
$\tan \theta$ = $\frac{AB}{BC}$

$\tan 60^{\circ}$ = $\frac{10}{x}$

$\sqrt{3}$ = $\frac{10}{x}$

$x$ = $\frac{10}{\sqrt{3}}$

$x$ = $\frac{10}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}$

= $\frac{17.32}{3}$

= 5.8 meter (approx)

Question 2: A ladder is leaning against the wall. Top of the ladder touches the wall at a point which is 4 meter high from the ground, while its bottom is 3 meter away from the base of wall. What is the length of the ladder?
Solution:
The following right triangle demonstrates the conditions given in question:Right Triangle Word Problem
Here, AC is the ladder and $\bigtriangleup ABC$ is the right triangle.

By Pythagorean theorem, In $\bigtriangleup ABC$:
$(AC)^{2}=(AB)^{2}+(BC)^{2}$
$(AC)^{2}=(4)^{2}+(3)^{2}$
$(AC)^{2}=16+9$
$(AC)^{2}=25$
$AC=\sqrt{25}$
$AC=5\ meter$

Right Triangle Problems

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Few right triangle problems are given below:

Solved Examples

Question 1: Determine whether the following sides make a right triangle:
(a) 8, 6, 10
(b) 6, 7, 8
Solution:
We know that in a right triangle, square of the longest side is equal to the sum of squares of other two sides.

(a)
Square of the longest side = $10^{2}=100$
Sum of squares of other two sides = $6^{2}+8^{2}$
= $36+64$ = 100
Square of the longest side = Sum of squares of other two sides.
Hence, these sides represent a right triangle.

(b)
Square of the longest side = $8^{2}=64$
Sum of squares of other two sides = $6^{2}+7^{2}$
= $36+49$ = 85
Square of the longest side $\neq $ Sum of squares of other two sides.
Hence, these sides do not represent a right triangle.

Question 2: Calculate the area and perimeter of a triangle whose length of base and height are 5 cm and 12 cm respectively.
Solution:
Area of triangle = $\frac{1}{2}$ x base x height

= $\frac{1}{2}$ x 5 x 12

= 30
Area of right triangle is 30$cm^{2}$

$(Hypotenuse)^{2}=5^{2}+12^{2}$
$(Hypotenuse)^{2}=25+144$
= 169
$Hypotenuse=\sqrt{169}$
$Hypotenuse=13 cm$

Perimeter of the right triangle = 5 + 12 + 13
= 30
Perimeter of the right triangle is 30 cm