Pythagoras Theorem
The Pythagoras Theorem states that the sum of the squares of the base and perpendicular is equal to the square of the hypotenuse in a right-angled triangle. Pythagoras' theorem states that if a triangle is right-angled (90 degrees), the square of the hypotenuse equals the sum of the squares of the other two sides.
Given a right angled triangle below,
the Pythagoras Theorem formula is given as,
(Hypotenuse)2=(Base)2+(Perpendicular)2
Pythagoras Theorem proof
The proof of the Pythagoras Theorem involves the concept of similarity of the triangle. It states that 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, right angled at Q.
We proof that, PR2=PQ2+QR2
Draw a perpendicular line QD meeting PR at D.
We know that, ΔRDQ∼ΔRQP
So,
QRRD=PRQR (Corresponding sides of similar triangles)
⟹QR2=RD×PR.......(1)
Also,
ΔQDP∼ΔRQP
So,
PQPD=PRPR (Corresponding sides of similar triangles)
So,
PQ2=PD×RP.......(2)
Adding equation (1) and (2), we get,
QR2+PQ2=(RD×PR)+(PD×PR)⟹QR2+PQ2=PR(RD+PD)
From the figure,
RD+PD=PR
Therefore,
PR2=PQ2+QR2 as required.
Applications of Pythagoras Theorem
- The Pythagoras Theorem is frequently used to determine the lengths of a right-angled triangle's sides.
- Architects employ the Pythagoras Theorem approach in the engineering and building professions.
- The Theorem is used to calculate the diagonal length of a rectangle, square, or other shape.
- It is also used in navigation to find the shortest route.
- Pythagoras Theorem is used in trigonometry to find the trigonometric ratios like cot,sin,cosec,cos,tan,sec.
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