Pythagoras Theorem

Presented By:

Mandeep Saluja (Science Mistress) Govt. Sen. Sec. School Patran

To enable the students to understand the Concept of

PYTHAGORAS THEOREM

In a right angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides.

A

B

C

GIVEN ABC is a right angled triangle in which

∠B = 90

0.

TO PROVE

AC = AB + BC 2

2

2

CONSTRUCTIONS From B , Draw BD ⊥ AC

PROOF In ∆ABD and ∆ ABC ; ∠ A = ∠ A ∠ ADB = ∠ ABC ( each =900 ) ∆ ABD Hence

∼

∆ ABC AB AD = AC AB

In similar triangles, the corresponding sides are proportional.

AB2 = AC x AD ------------ (1)

Simlarly in ∆BDc and ∆ ABC ; ∠ C = ∠ C ∠ BDC = ∠ ABC ( each =900 ) ∆ BDC ∼ ∆ ABC Hence,

BC DC ⇒ = AC BC

BC2 = AC x DC ----------- ( 2)

By adding (1) and (2),We get AB2 + BC2 = AC x AD + AC x DC = AC ( AD + DC) = AC x AC [...AD+DC=AC] = AC2 Hence,

AB + BC = AC 2

2

Hence Proved

2

• If two sides of a triangle are given third can be found with the help of Pythagoras Theorem

By Pythagoras Theorem, AB2 + BC2 = AC2 32 + 4 2 = X2 9 + 16 = X2 25 = X2 5 = X

• We can find whether the given triangle is right angle or not if sides AB=5 cm, BC=12 cm, AC=13 cm

By Pythagoras Theorem, AB2 + BC2 = AC2 52 + 12 2 = 13 2 25 + 144 = 169 169 = 169 Hence given triangle is right angled.

We can fnd the height of a object with the help of pythagorus theorem. Example: The lower edge of a ladder of length 17m is at a distance of 8m from the wall. Upper edge is laying on a window. Find out the distance of window from the floor. • By Pythagoras Theorem,

AB2 + BC2 = AC2 AB2 + 82 = 172 AB2 + 64 = 289 AB2 = 289 - 64 AB2 = 225 AB2 = 25

Height of tree is 15 m.