Pythagorean Theorem Practice
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Pythagorean Theorem Practice
The Pythagorean theorem is related to the study of sides of a right angled triangle. It is also
called as pythagoras theorem. The pythagorean theorem states that, In a right triangle, (length
of the hypotenuse)2 = {(1st side)2 + (2nd side)2}.
In a right angled triangle, there are three sides: hypotenuse, perpendicular and base. The
base and the perpendicular make an angle of 90 degree with eachother. So, according to
pythagorean theorem: (Hypotenuse)2 = (Perpendicular)2 + (Base)2
In the above figure1,
c2 = a2 + b2
Therefore, Hypotenuse (c) = (a2 + b2)
Pythagorus Theorem ProofBack to Top
From the above figure 2, ABC is a right angled triangle at angle C.
From C put a perpendicular to AB at H.
Now consider the two triangles ABC and ACH, these two triangles are similar to each
other because of AA similarity. This is because both the triangle have a right angle and one
common angle at A.
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So by these similarity,
ac = ea and bc = db
a2 = c*e and b2 = c*d
Sum the a2 and b2, we get
a2 + b2 = c*e + c*d
a2 + b2 = c(e + d)
a2 + b2 = c2 (since e + d = c)
Hence Proved.
Euclid Proof of Pythagorean Theorem
According to Euclid, if the triangle had a right angle (90 degree), the area of the square
formed with hypotenuse as the side wil be equal to the sum of the area of the squares formed
with the other two sides as the side of the squares. From the above figure 3, the sum of the
area covered by the two smal squares is equal to the area of the third square. Here, a2 is the
area of the square ABDE, b2 is the area of the square BCFG and c2 is the area of the square
ACHI.
Therefore, a2 + b2 = c2
Hence Proved.
Pythagorean Theorem Example Problems
Below are example problems on Pythagorean theorem
Example Problem 1: In a right triangle, the hypotenuse is 5 cm and the perpendicular is 4 cm.
Find the length of the base of the triangle?
Solution: By using Pythagoras theorem, h2 =p2 + b2
52 = 42 + b2, 25 = 16 + b2, 9 = b2, b = 3 Base is 3 cm
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Definition of Circle
Definition of Circle
Circle is defined as the set of points that is at an equal distant from the centre of the
circle.There are a number of terminologies involved in a Circle. Some of them are as fol ows:
Centre: The predetermined point from which the surface of the circle is at an equidistant is
called the centre of a circle.
Radius: The constant distance from the centre to a point on the surface of the circle is called
its radius .
Circumference: The boundary of a circle is cal ed its circumference.
Chord: A line segment whose end points is present on the circumference of a circle is cal ed a
chord .
Diameter: A chord crossing through the midpoint of a circle is cal ed its diameter.
Circle Formulas
Diameter of a Circle: Diameter = 2 X Radius
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Radius of a Circle: Radius(R) = Diameter / 2
Area of a Circle: Area = pi X R2
Circumference of a Circle: Circumference = 2 X pi X R
Circle Theorem
Theorem 1: A perpendicular from the centre of a circle to a chord bisects the chord.
Given : AB is a chord in a circle with centre O. OC
AB.
To prove: The point C bisects the chord AB.
Construction: Join OA and OB
Proof: In triangles OAC and OBC,
mOCA = m O
CB = 90 (Given)
OA = OB (Radii)
OC = OC (common side)
OAC = OBC (RHS)
CA = CB (corresponding sides)
The point C bisects the chord AB.
Hence the theorem is proved.
Theorem 2: AB and CD are equal chords of a circle whose centre is O. OM
AB and ON
CD. Prove that m O
MN = m O
NM.
Given : In a circle with centre O chords AB and CD are equal
OM
AB, ON
CD (Fig.6.11).
To prove : O
MN = O
NM
Proof : AB = CD (given)
OM
AB (given); ON
CD (given)
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ThankYou
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