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Math Help: Introduction to the PythagoreanTheorem

written by: Suvo • edited by: Trent Lorcher • updated: 6/6/2012

If you've taken any geometry or trigonometry you've probably become familiar with the Pythagorean Theorem. Get some help understanding the concept behind the theorem with these examples.

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    Around 2530 years ago, Pythagoras first created the Pythagorean Theorem. A simple Pythagorean Theorem proof is making a pyramid with a perfect square or rectangular base.

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    The Theorem

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

    See the attached snap of the triangle below (Click on it to enlarge):

    Pythagorus Theorem Explanation 

    So, as per Pythagoras theorem we can write that:

    (BC)2 = (AB)2 + (AC)2

    If we consider the side AB as a, AC as c and BC as b then we can write the Pythagorean Theorem as

    b2=a2+ c2

    Now you are in a position to calculate the length of any one side of a right angled triangle, if the lengths of the other two sides are given.

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    Practice Problems

    Solve the following examples. Assume c is the hypotenuse of a right angled triangle and a, b are the other two sides of the same triangle:

    a = 3; b = 4; c=?

    a=5; c= 13; b=?

    a=8; b=15; c=?

    a= 9; c=41; b=?

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    Pythagorean Theorem Proof

    Pythagoras K12 2 

    Pythagoras' Theorem has more than 300 proofs. The simplest proof of the theorem is based on the similar triangles concept:

    • Take the triangle ABC with AB=a, AC=c and BC=b.
    • Drop a line from A to D which is perpendicular to BC.
    • Consider AD=d.
    • Triangle ACD is similar to the triangle ABC as:

    Angle ADC = Angle CAB = 90 degree

    Angle ACD = Angle ACB

    Side AC is common for both the triangle

    So, we can write from similar triangle principle:

    c / b = DC / c

    c2 = b X DC …………….eqn.1

    • Again, triangle ABD and triangle ABC are similar because:

    Angle ADB = Angle CAB = 90 degree

    Angle ABD = Angle ABC

    Side AB is common for both triangles.

    So, we can write:

    a / b = BD / a

    a2 = b X BD …………….eqn.2

    • From eqn.1 and eqn.2 we can write:

    c2 +a2 = (b X DC) + (b X BD)

    = b X (DC + BD)

    = b X BC(as, BC consists of DC and BD)

    = b X b(as, we already assume BC=b)

    = b2

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    Real Life Examples

    • Making a perfect rectangular basketball and volleyball court.
    • Measuring the height of ramp.
    • Calculating distance between two points if co-ordinates of the points are given.
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    Pythagoras Triples

    Pythagoras triples are sets of three integer numbers which follows Pythagoras' Theorem. For example, take 3, 4, 5.

    Remember the Pythagoras theorem (b2=a2+ c2).

    Now, if you take a=3, c=4 then from the theorem b will be equal to 5. There are many such sets of integers like this: 5, 12,13; 17, 24, 25; 9, 40, 41 etc.

    People have been using these triples even before human beings learned to write.