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Written byS H W E T H A B . R | Last Modified 10-01-2022 | 0 C O M M E N T S
Pythagoras Theorem: Formula, Examples, Proof
The Pythagorean Theorem describes how the sides of a right-angled triangle are
related. The Pythagorean Theorem is also another name for it. In a right-angled
triangle, the Pythagoras Theorem is frequently used to determine the length of an
unknown side. The theorem outlines the relationship between the base,
perpendicular, and hypotenuse of a right-angled triangle. The converse of
Pythagoras Theorem is used to determine whether or not a triangle is right-angled.
Pythagoras Theorem is an important topic in Maths.
Generally, many students find Maths to be a tough subject. It is very important to
learn these concepts and with regular practice, they can master the subject. These
basic concepts are part of the bigger chapter in higher classes. That is why it is
crucial to allocate more time for Mathematics and learn essential concepts like the
Pythagoras theorem. Continue reading to know more.
Table of Contents
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Statement of Pythagoras Theorem
o Pythagoras Theorem Formula
o Basic Terms of Pythagoras Theorem
o Pythagorean Triplets
o Pythagoras Theorem Proof
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Converse of Pythagoras Theorem and Its Proof
Applications of Pythagoras Theorem
Solved Examples
Summary
FAQs
Statement of Pythagoras Theorem
The Pythagoras Theorem states that in a right-angled triangle the sum of the square
of the base and square of the perpendicular is equal to the square of the
hypotenuse. If a triangle is right-angled (90 degrees), the square of the hypotenuse
equals the sum of the squares of the other two sides, according to Pythagoras’
theorem. Look at the triangle ABC below, where BC2 = AB2 + AC2. The base is AB,
the altitude (height) is AC, and the hypotenuse is BC.
(Hypotenuse)2=(Base)2+(Perpendicular)2(Hypotenuse)2=(Base)2+(Perpendic
ular)2
Pythagoras Theorem Formula
(Hypotenuse)2=(Base)2+(Perpendicular)2(Hypotenuse)2=(Base)2+(Perpendic
ular)2
Where, b=b= Base, a=a= Perpendicular, c=c= Hypotenuse
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Basic Terms of Pythagoras Theorem
The basic terms used in the Pythagoras Theorem are base, perpendicular (altitude),
and hypotenuse. The two sides of a right-angled triangle that forms the right angle
are known as base and perpendicular, and the side which is opposite to the right
angle is known as the hypotenuse.
Base
LEARN ABOUT PYTHAGORAS THEOREM CONCEPT
Base is the side in a right-angled triangle which is opposite to the angle
Perpendicular (altitude or height)
Perpendicular is the side which is opposite to the angle R.R.
Hypotenuse
Hypotenuse is the side which is opposite to the right angle i.e., angle
P.P.
Q.Q.
Pythagorean Triplets
Three positive integers a,ba,b and cc are called Pythagorean triplets
if c2=a2+b2,c2=a2+b2, the triple is commonly written (a,b,c)(a,b,c) i.e., in increasing
order of a,ba,b and c.c.
Examples for Pythagorean triplets
are (3,4,5),(5,12,13),(7,24,25),(8,15,17),(9,40,41).(3,4,5),(5,12,13),(7,24,25),(
8,15,17),(9,40,41).
Pythagoras Theorem Proof
The proof of the Pythagoras Theorem is very interesting. It involves the concept of
similarity of the triangle.
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the
squares of the other two sides.
Given: A right-angled triangle PQR,PQR, right angled at Q.Q.
To prove: PR2=PQ2+QR2PR2=PQ2+QR2
Construction: Draw a perpendicular line QDQD meeting PRPR at D.D.
PRACTICE EXAM QUESTIONS
Proof: we know that ΔRDQ∼ΔRQPΔRDQ∼ΔRQP
So, RDQR=QRPRRDQR=QRPR (Corresponding sides of similar triangles)
⇒QR2=RD×PR⇒QR2=RD×PR —(i)(i)
Also, ΔQDP∼ΔRQPΔQDP∼ΔRQP
So, PDPQ=PQPRPDPQ=PQPR (Corresponding sides of similar triangles)
⇒PQ2=PD×RP⇒PQ2=PD×RP —(ii)(ii)
Adding the equation (i)(i) and (ii)(ii) we get,
QR2+PQ2=RD×PR+PD×PRQR2+PQ2=RD×PR+PD×PR
⇒QR2+PQ2=PR(RD+PD)⇒QR2+PQ2=PR(RD+PD)
From the figure, RD+PD=PRRD+PD=PR
From the figure, PR2=PQ2+QR2PR2=PQ2+QR2
Hence, the Pythagoras Theorem is proved.
Converse of Pythagoras Theorem and Its Proof
In a triangle, if the square of one side is equal to the sum of the squares of the other
two sides, then the angle opposite the first side is a right angle.
Given: In ΔXYZ,XY2+YZ2=XZ2ΔXYZ,XY2+YZ2=XZ2
∠XYZ=90∘∠XYZ=90∘
Construction: – A triangle PQRPQR is constructed such that
PQ=XY,QR=YZ,∠PQR=90∘PQ=XY,QR=YZ,∠PQR=90∘
Proof:In ΔPQR,∠Q=90∘ΔPQR,∠Q=90∘
PR2=PQ2+QR2PR2=PQ2+QR2 [Pythagoras Theorem]
Or PR2=XY2+YZ2PR2=XY2+YZ2 —(i)(i)
[PQ=XY,QR=YZ][PQ=XY,QR=YZ]
Therefore, XZ2=PR2XZ2=PR2 [From equation (i)(i) and (ii)(ii)
Or XZ=PRXZ=PR
Or ΔXYZ≅ΔPQRΔXYZ≅ΔPQR [SSSSSS congruency rule]
Therefore ∠Y=∠Q=90∘∠Y=∠Q=90∘ [CPCT]
Hence, ∠XYZ=90∘∠XYZ=90∘
The converse of Pythagoras Theorem is proved.
ATTEMPT MOCK TESTS
Applications of Pythagoras Theorem
Some of the applications of the Pythagoras Theorem are
1. Pythagoras Theorem is commonly used to find the lengths of sides of a rightangled triangle.
2. The Theorem is used to find the length of diagonal of a rectangle, square, etc.
3. Pythagoras Theorem is used in trigonometry to find the trigonometric ratios
like sin,cos,tan,cosec,sec,cot.sin,cos,tan,cos⁡ec,sec,cot..
4. Pythagoras Theorem is used in security cameras for face recognition.
5. The technique of Pythagoras Theorem is used by architects for engineering and
construction fields.
6. The Pythagoras Theorem is applied in surveying the mountains.
7. It is also used in navigation to find the shortest route.
8. By using the Pythagoras Theorem, we can derive formula for base, perpendicular
and hypotenuse.
9. Painters use ladders to paint on high buildings with the help of Pythagoras
Theorem.
10. The Theorem is used to calculate the steepness of slopes of hills or mountains.
11. The converse of Pythagoras Theorem is used to check whether a triangle is a
right triangle or not