Answer to Question #287314 in Trigonometry for Mum

Question

Explain the Pythagorean Theorem, its proofs and applications.

Answer 1

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, ${PR^2}={PQ^2}+{QR^2}$

Draw a perpendicular line QD meeting PR at D.

We know that, $\Delta RDQ\sim\Delta RQP$

So,

${RD\over QR}={QR\over PR}$ (Corresponding sides of similar triangles)

$\implies{QR^2}=RD\times PR.......(1)$

Also,

$\Delta QDP\sim\Delta RQP$

So,

${PD\over PQ}={PR\over PR}$ (Corresponding sides of similar triangles)

So,

$PQ^2=PD\times RP.......(2)$

Adding equation (1) and (2), we get,

$QR^2+PQ^2=(RD\times PR)+(PD\times PR)\\ \implies QR^2+PQ^2=PR(RD+PD)$

From the figure,

$RD+PD=PR$

Therefore,

$PR^2=PQ^2+QR^2$ as required.

Applications of Pythagoras Theorem

1. The Pythagoras Theorem is frequently used to determine the lengths of a right-angled triangle's sides.
2. Architects employ the Pythagoras Theorem approach in the engineering and building professions.
3. The Theorem is used to calculate the diagonal length of a rectangle, square, or other shape.
4. It is also used in navigation to find the shortest route.
5. Pythagoras Theorem is used in trigonometry to find the trigonometric ratios like $cot,sin,cosec,cos,tan,sec.$