Right Angle Triangle Theorem Formula Proof and Solved Examples
The relationship between a perpendicular, hypotenuse, and base of a triangle is stated by the Pythagoras Theorem. This can also be called the right-angled triangle theorem. According to the Pythagoras theorem, if the sum of the square of perpendicular and the square of the base is equal to the square of the hypotenuse, then it is known as a right-angled triangle. It can be expressed as follows:
If Hypotenuse2 = Perpendicular2 + Base2, then angle = 900
The side that is opposite to the angle is known as the hypotenuse, the adjacent side to the right angle, and is connected to the hypotenuse is the perpendicular and the base is the adjacent side of the right angle. The base is also called as the legs of the hypotenuse. The length of any side of the triangle can be found if we have two known values of the triangle. In this triangle, one angle is equal to 900 and the other two angles are equal to the sum of 900. You can always change the base and the perpendicular sides depending on how we consider the acute angle. The two sides of the triangle, i.e., base, hypotenuse, and the perpendicular are considered to be Pythagoras triplets and it is known as Pythagorean triangle if all the sides are integers.
Pythagoras Theorem Proof
Theorem: In a right-angled triangle, if the sum of the square of two sides is equal to the square of one side, then the right angle is the angle that is opposite to the first side.
Proof of right-angle triangle.
To prove ∠Q = 90.
Proof: We have a triangle in which PR2 = PQ2 + QR2
We need to prove that ∠Q = 90.
To prove the above condition, we need to first construct a triangle ΔLMN such that M is the right angle and is equal to 900
LM2 = LN2+ NM2 (According to Pythagoras theorem, as ∠N = 900)
or, LM2 =PQ2+ QR2 ( By construction ) …… (1)
We know that;
AC2 = PQ2 + QR2 ( Which is given ) …………(2)
So, AC = LM Fromequation(1)and(2)
Now, in Δ ABC and Δ LMR,
PQ = LM ( By construction )
QR = MN ( By construction )
PR = LN Provedabove
So, Δ PQR ≅ Δ LMN ( By SSS congruence )
Therefore, ∠Q = ∠M ( CPCT )
But, ∠M = 900 ( By construction )
So, ∠Q = 900
Hence the theorem is proved. Pythagoras' theorem can be proven in a variety of ways. The algebraic method and the similar triangles method are two of the most common and widely used methods.
Algebraic Method
The algebraic technique can be used to derive the proof of Pythagoras' theorem.
Let’s assume a square PQRS and arrange four congruent right triangles in the given square, whose side is a + b.
The 4-right triangles have 'b' as the base, 'a' as the height and, 'c' as the hypotenuse.
The 4 triangles form the inner square WXYZ, with 'c' as the four sides.
The area of the square WXYZ is c2 when the four triangles are arranged in a square.
The area of the square PQRS with side (a + b) = Area of 4 triangles + Area of the square WXYZ with side 'c'.
This means (a + b)2 = (4 × 1/2 × (a × b)) + c2.
This leads to a2 + b2 + 2ab = 2ab + c2.
Therefore, a2 + b2 = c2.
Hence, proved.
Similar Triangle Method
Two triangles are considered to be comparable if their respective angles are the same size and their corresponding sides have the same proportion. Also, if the angles have the same measure, we may say that the corresponding sides will have the same ratio using the sine law. As a result, equivalent angles in identical triangles result in equal side length ratios.
Sine, Cosine, and Tangent for Right Triangles
Angle functions such as sine, cosine, and tangent are the most useful in right triangle calculations. These three functions have the following relationships with right triangles: In the case of a right triangle,
The length of the leg opposite the angle divided by the length of the hypotenuse is the sine of a non-right angle.
A non-right angle's cosine is equal to the length of the adjacent leg divided by the length of the hypotenuse.
A non-right angle's tangent is equal to the length of the leg opposite that angle divided by the length of the leg adjacent to it.
Right Angle Triangle Formula
The formula of the right-angled triangle goes by:
( Hypotenuse )2 = ( Opposite )2 + ( Adjacent )2
If p, q and r are hypotenuses, adjacent, and the opposite of a right-angles triangle, then:
p2 = r2 + q2
or
\[ p = \sqrt{r^{2}+q^{2}} \]
This means, the root of the sum of the squares of the base and the perpendicular.
Half of the product of the base and the perpendicular is equal to the area of the triangle.
Right Angle Triangle Formula is:
A = 1/2 x base x height
Here,
B = base
H = height of the triangle
Solved Examples
Question 1: The angle ∠JLK = 900 and LP is perpendicular to K. Prove that KJ2/ JL2 = KP / JP.
Solution: We can see that;
△ JJL ~ △ JLK
According to the property of similar triangles we have:
JL / JK = JP / JL
or it can also be written as, JL2 = JK . JP ……….(1)
Similarly, we have △ KPL ~ KLJ
So we have, KP / KL = KL / KJ
Or it can also be written as;
KL2 = KJ . KP ……….. (2)
By dividing the equations (2) by (1) we can deduce that:
LL2 / JL2 = ( LJ . LP ) / ( JL . JP ) = LP / JP
Hence, it proved. Let’s study some right-angle triangle examples.
Question 2: If the length of the hypotenuse and the base is 6 cm and 5 cm of the right triangle. Find the height of the triangle.
Solution: We know the formula of the right-angle triangle. It goes by:
( Hypotenuse )2 = ( Opposite )2 + ( Adjacent )2
( 6 )2 = ( 5 )2 + ( Perpendicular )2
36 = 25 + ( P )2
36 - 25 = ( P )2
P = \[\sqrt{9}\] = 3 cm
Hence, the height of the perpendicular is 3 cm.
Question 3: If the length of the hypotenuse and the base is 10 cm and 6 cm of the right triangle. Find the height of the triangle.
Solution: We know the formula of the right-angle triangle. It goes by:
( Hypothenuse )2 = ( Opposite )2 + ( Adjacent )2
( 10 )2 = ( 6 )2 + ( Perpendicular )2
100 = 36 + ( P )2
100 - 36 = ( P )2
P = \[\sqrt{64}\]
= 8 cm
Hence, the height of perpendicular is 8 cm.
FAQs on Understanding the Right Angle Triangle Theorem
1. What is the Right Angle Triangle Theorem?
The Right Angle Triangle Theorem, commonly known as the Pythagorean Theorem, states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
It is written as:
a² + b² = c²
Where:
- a and b are the legs (perpendicular sides)
- c is the hypotenuse (longest side)
2. What is the formula for a right-angled triangle?
The formula for a right-angled triangle is a² + b² = c², where c is the hypotenuse.
To use the formula:
- Square the two shorter sides (a and b)
- Add the squares
- Take the square root to find c if needed
3. How do you find the hypotenuse of a right triangle?
To find the hypotenuse of a right triangle, use the formula c = √(a² + b²).
Steps:
- Square both legs
- Add the squares
- Take the square root of the result
c = √(3² + 4²) = √(9 + 16) = √25 = 5.
4. How do you find a missing side in a right triangle?
To find a missing side in a right triangle, rearrange the Pythagorean Theorem depending on the unknown side.
If a leg is missing:
a = √(c² − b²)
Steps:
- Square the hypotenuse
- Subtract the square of the known leg
- Take the square root
a = √(100 − 36) = √64 = 8.
5. Why does the Pythagorean Theorem only work for right triangles?
The Pythagorean Theorem only works for right triangles because it depends on the presence of a 90° angle between the two legs.
The relationship a² + b² = c² is derived specifically from the geometry of right angles. In non-right triangles, different formulas like the Law of Cosines are used instead.
6. What is a Pythagorean triple?
A Pythagorean triple is a set of three whole numbers that satisfy a² + b² = c².
Common examples include:
- 3, 4, 5
- 5, 12, 13
- 8, 15, 17
7. How do you check if a triangle is a right triangle?
To check if a triangle is a right triangle, verify whether a² + b² = c² using the longest side as c.
Steps:
- Identify the longest side
- Square all three sides
- Check if the sum of the two smaller squares equals the largest square
8. What is the hypotenuse in a right-angled triangle?
The hypotenuse is the longest side of a right-angled triangle and lies opposite the 90° angle.
Key properties:
- It is always the largest side
- It is represented by c in the formula a² + b² = c²
- It connects the endpoints of the two perpendicular sides
9. Can you give a real-life example of the Right Angle Triangle Theorem?
The Right Angle Triangle Theorem is used in real life to calculate distances, heights, and construction measurements.
Example: A ladder placed 6 m from a wall reaches 8 m up the wall. The ladder length is:
c = √(6² + 8²) = √(36 + 64) = √100 = 10 m.
This application is common in engineering, architecture, and navigation.
10. What is the difference between the Pythagorean Theorem and the Law of Cosines?
The Pythagorean Theorem applies only to right triangles, while the Law of Cosines works for all types of triangles.
Pythagorean Theorem:
a² + b² = c² (only when angle = 90°)
Law of Cosines:
c² = a² + b² − 2ab cos(C)
When angle C = 90°, cos(90°) = 0, and the Law of Cosines reduces to the Pythagorean formula.
