Pythagoras Theorem
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 Right Angle Triangle Theorem
1. How to Prove a triangle is Right Angled?
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 also says that, 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. It can be expressed as follows:
(Hypotenuse)2 = (Perpendicular)2 + (Base)2, then angle = 900
If in a triangle PQR, (PR)2=(PQ)2+(QR)2, then angle Q = 900 (right angle at Q)
2. Explain the derivation of the Pythagoras theorem?
Let’s take a right-angled triangle ABC.
Right-angled at B.
A perpendicular BD is to be drawn to meet AC at D.
In △ABD and △ACB,
∠A = ∠A (common)
∠ADB = ∠ABC (both are right angles)
Thus, △ABD ∼ △ACB (by AA similarity criterion)
Similarly, we can prove △BCD ∼ △ACB.
Thus △ABD ∼ △ACB,
Hence, AD/AB = AB/AC.
It is clear that AD × AC = AB2.
Similarly, △BCD ∼ △ACB.
Hence,CD/BC = BC/AC.
Also, CD × AC = BC2.
Adding these 2 equations:
AB2 + BC2 = (AD × AC) + (CD × AC)
AB2 + BC2 =AC(AD +DC)
AB2 + BC2 =AC2
Hence, proved.
3. What are the applications of the Pythagoras theorem?
The Pythagoras theorem is used in everyday life. Here are several Pythagoras applications.
A Pythagorean theorem is a well-known tool for determining unknown dimensions. It is simple to calculate a sector's diameter from its length or breadth. It is mostly used in engineering in two dimensions.
Face recognition in security cameras uses the Pythagorean theorem, which measures the distance between the camera and the person being recognised.
The Pythagoras concept is used in interior design and building architecture.
Those going by sea utilize this method to find the shortest distance and route.
3. Who was Pythagoras?
Pythagoras of Samos (c. 570 – 495 BC) was a well-known Greek mathematician and philosopher. He is best known for proving the important Pythagorean theorem about right-angle triangles. He found the Pythagoreans, a group of mathematicians who worshipped numbers and lived like monks. He had an impact on Plato and a significant influence on mathematics. His theories continue to be used in mathematics today. He was regarded as one of the greatest thinkers of his generation.
4. What are Pythagorean Triples?
Pythagorean triples or triplets are three whole numbers that fit the equation a2+b2=c2, where a,b,c are said to be Pythagorean triplets.
Eg: A well-known example is a triangle with sides of 3, 4, and 5.
If a=3 and b=4,
then 32+42=52
So, c=5
Since 9+16=25
This can also be shown as \[\sqrt{3^2+4^2}\] =25.
All multiples of 3, 4, and 5 work with the three-four-five triangle. Other examples of Pythagorean triples are (6, 8, 10); (7,24,25); (5,12,13)....
