Pythagoras Theorem

The Pythagoras theorem is attributed to a Greek mathematician Pythagoras and his group, the Brotherhood of the Pythagoras over 2000 years ago. Their contribution in mathematics developed an algebraic method applied in geometry.

Pythagoras Theorem Pythagoras Theorem

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Contents
Table of contents

    History of the Pythagoras theorem

    What does the Pythagoras theorem explain?

    The Pythagoras theorem tells us that if the sides of a right angled-triangle or right triangle are squares, the area of the biggest square is the same as the sum of the area of the two smaller squares.

    Pythagoras Theorem An image illustrating the relationship between squares and a right triangle StudySmarterAn image illustrating the relationship between squares and a right triangle, Njoku - StudySmarter Originals

    Pythagoras Theorem An image using squares to prove the Pythagoras theorem StudySmarterAn image using squares to prove the Pythagoras theorem, Njoku - StudySmarter Originals

    So, the area of the biggest square equals the sum of the areas of both smaller squares;

    a2=b2+c2

    This is what the Pythagoras theorem explains.

    Therefore, the Pythagoras theorem states that when a triangle has one of it angles equal to 90 degrees, then the square of the longest side equals the sum of the squares of the two other sides.

    The longest side is called the hypotenuse, the vertical side is called the opposite and the horizontal side is called the adjacent.

    Pythagoras Theorem An illustration on the sides of a right triangle StudySmarter An illustration on the sides of a right triangle, Njoku - StudySmarter Originals

    So the formula of the Pythagoras theorem is;

    hypotenuse2=opposite2+adjacent2

    Find the value of x in the figure below;

    Using Pythagoras theorem, we can see that our opposite and adjacent is given but out hypotenuse is given as x. Thus;

    hypotenuse2=opposite2+adjacent2x2=52+122x2=25+144x2=169

    Find the square root of both sides

    x=13cm

    If a right angled triangle has equal dimension in two of it sides and the longest side measures 8cm. Find the other sides.

    Solution.

    From the question our hypotenuse is given as 8cm. However, the opposite and adjacent are not given. Also, we are told opposite = adjacent.

    let the adjacent = y; that means opposite = y. Therefore:

    Using pythagoras theorem, hypotenuse2=opposite2+adjacent282=y2+y264=2y2

    Divide both sides by 2

    642=2y2232=y2

    Find the square root of both sides of the equation

    32=y2y=42cm

    So the opposite is 4√2cm and the adjacent is 4√2cm.

    If sinØ = 2/5, Find cosØ and tanØ.

    Solution

    SinØ = oppositehypotenuse=25

    This means that the opposite is 2 and the adjacent is 5. Meanwhile, we need to find the adjacent:hypotenuse2=opposite2+adjacent252=22+adjacent29=4+adjacent2

    Subtract 4 from both sides of the equation.

    9-4=4+adjacent2-45=adjacent2

    Take the square roots.

    adjacent=5

    Now, we have values for all sides.

    cos=adjacenthypotenusecos=55Tan=oppositeadjacenttan=25

    Rationalize by multiplying the denominator and numerator by √5.

    tan=2×55×5tan=255

    What is the Pythagorean triple?

    A Pythagorean triple consists of 3 sets of numbers which prove correctly the Pythagoras theorem. This means that the square of the highest number among this numbers must be equal to the sum of the squares of the other two numbers in the set.

    Determine if the following is a Pythagorean triple.

    1. 7, 12 and 5

    2. 8, 15 and 17

    Solution

    1. To confirm if the series 7, 12 and 5 are Pythagorean triples, take the square of the largest number.

    The largest number is 12 and its square is 144.

    You should sum the squares of the other two numbers in the series.

    the square of 7 is 49

    the square of 5 is 25

    49+25 = 74

    1447412272+52

    This means that the series 7, 12 and 5 is not a Pythagorean triple.

    2. To confirm if the series 8, 15 and 17 are Pythagorean triples, take the square of the largest number.

    The largest number is 17 and its square is 289.

    You should sum the squares of the other two numbers in the series.

    the square of 8 is 64

    the square of 15 is 225

    64+225 = 289

    289=289172=82+152

    This proves that the set 8, 15 and 17 is a Pythagorean triple.

    Pythagoras Theorem - Key takeaways

    • The Pythagoras theorem tells us that if the sides of a right angled-triangle or right triangle are squares, the area of the biggest square is the same as the sum of the area of the two smaller square.
    • The longest side is called the hypotenuse, the vertical side is called the opposite and the horizontal side is called the adjacent.
    • The formula of the Pythagoras theorem is; hypotenuse2=opposite2+adjacent2

    • A Pythagorean triple consists of 3 sets of numbers which prove correctly the Pythagoras theorem.

    Frequently Asked Questions about Pythagoras Theorem

    What is Pythagoras theorem? 

    This is a principle that enables us to apply algebra in geometry by the determination of the sides of a right triangle.

    What is the hypotenuse in Pythagorean theorem? 

    The hypothenuse in Pythagorean theorem is the longest side of the right triangle.

    How to do Pythagorean theorem? 

    Pythagorean theorem is applied when finding one of the sides of a right triangle when two other sides are given.

    What is the formula of the Pythagoras theorem?

    The formula of the Pythagoras theorem is; hypothenuse square = opposite square + adjacent square

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