What is a Triangle?
The triangle is a closed geometric shape with three sides, three vertices and three angles. In our everyday life we come across various objects that are triangular in shape such as a slice of pizza, a cloth hanger, traffic signs and so on.
Properties of a Triangle
- The sum of all interior angles in a triangle is 180°.
- The sum of length of any two sides is always greater than the length of the third side.
- The difference between the length of two sides is always less than the length of the third side.
Types of Triangles
Based on the sides and the angles of a triangle, they are classified into six types;
What is a Right Angle Triangle?
A triangle with one angle as 90°, is called a right angle triangle. The remaining two angles add up to 90°. The sides that include the right angle are known as the legs of the triangle. The third side is called the hypotenuse, which is the longest side.
Note: The hypotenuse is the side opposite to the right angle.
In the above figure,
∠R = 90°,
QR & PR are forming the legs, and PQ is the hypotenuse of the right angled triangle.
Pythagorean Theorem
The three sides of a right angled triangle are related to one another and this relationship is well explained by the Pythagorean theorem.
The Pythagorean theorem states that, in a right angled triangle the sum of the squares of the lengths of the legs is equal to the square of the length of hypotenuse.
So as per the Pythagorean theorem, in the right triangle ABC:
\( (AB)^2~+~(BC)^2~=~(AC)^2 \)
Formula for the Area and Perimeter of a Right Angled Triangle
Area of right angle triangle = \(=~\frac{1}{2}~\times~\text{base}~\times~\text{height} \) \(=~\frac{1}{2}~bh \)
Perimeter = Sum of lengths of all the sides
Note: The legs of a right angled triangle are its base and height.
Properties of Right Angle Triangle
Let us look at a few properties of a right angle triangle;
- One angle is always a right angle, that is 90°.
- The sum of the other two interior angles adds up to 90°.
- The side opposite to the right angle is called hypotenuse and that is the longest side of the triangle.
- The remaining two sides forming the right angle are called the legs.
Rapid Recall
Solved Examples on Right Angle Triangle
Example 1: Find the measure of the missing angle.
Solution:
We have been given a right angled triangle with an unknown angle x.
The other two angles are 90° and 25°.
We know sum of angles of a triangle is 180°, that is,
25° + 90° + x = 180°
115° + x = 180° [Add]
x = 180° – 115° [Subtract 115 from both sides]
x = 65°
Therefore the missing angle is 65°.
Example 2: Find the area of a triangle whose height and base is 20 units and 13 units respectively.
Solution:
Area of triangle = \(\frac{1}{2}~\times~\text{base}~\times~\text{height} \)
= \(\frac{1}{2}~\times~13~\times~20 \) [Substitute the value of base and height]
= 130 square units [Simplify]
Therefore, the area of the triangle is 130 square units.
Example 3: For a triangle with three sides measuring 5 units, 12 units, and 13 units, check whether the triangle is a right angle triangle or not.
Solution:
By Pythagorean theorem, \(a^2~+~b^2~=~c^2 \),
where c is the longest side of the triangle.
Let a = 5, b = 12 and c = 13
\(a^2~+~b^2~=~5^2~+~12^2~=~25~+~144~=~169 \)
\(c^2~=~13^2~=~169 \)
∴ \(a^2~+~b^2~=~c^2 \)
Since the given length of sides satisfies the Pythagorean theorem, so the triangle formed by these sides is a right angle triangle
Example 4: John and Max were standing at a point on the soccer field. John took 9 steps to the North and Max took 12 steps to the East. What is the distance in steps between them?
Solution:
John took 9 steps to the North and Max took 12 steps to the East, so the angle between their paths is 90°.
To find the distance between John and Max we can use the Pythagorean theorem.
The distance between them is denoted by \(d \)
\(d^2~=~9^2~+~12^2 \) [By Pythagorean theorem]
\(\Rightarrow ~d^2~=~81~+~144 \) [Simplify]
\(\Rightarrow ~d^2~=~225 \)
\(\Rightarrow ~\sqrt{d^2}~=~\sqrt{225} \) [Take square root on both sides]
\(\therefore ~d~=~15 \) units
Hence the distance ‘\(d \)’ between John and Max is 15 steps.
A triangle with all the sides of different length is called a scalene triangle.
The angle sum property of a triangle states that the sum of all the interior angles in a triangle adds up to 180°.
A triangle cannot have two obtuse angles.
Pythagorean triples are the integers which satisfies the Pythagorean theorem, that is, \(a^2~+~b^2~=~c^2 \), where a and b are the legs and c is hypotenuse.
For example: 3, 4 and 5 are Pythagorean triples.
\(3^2~+~4^2~=~5^2 \)
LHS = \(3^2~+~4^2~=~9~+~16~=~25 \)
RHS = \(5^2 ~=~25\)
\(\Rightarrow \) LHS = RHS
If one angle is 90° and the other two angles are 45° each, then the triangle is called an ‘Isosceles Right Angle Triangle’, where the sides forming the right angle are equal in length.