Prove Pythagorean Theorem
Grade 8 Math Worksheets
The Pythagorean Theorem states that the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.
Table of Contents:
- Pythagorean Theorem
- Proof of the Pythagorean Theorem
- Solved Example
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Prove Pythagorean Theorem - Grade 8 Math Worksheet PDF
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The Pythagorean Theorem is a mathematical formula that relates to the sides of a right triangle. It states that the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. The theorem can be written as:
a^2 + b^2 = c^2
Where c is the length of the hypotenuse, and a and b are the lengths of the other two sides. The Pythagorean Theorem can be used to find the length of any side of a right triangle, as long as the lengths of the other two sides are known.
The Pythagorean Theorem is named after the ancient Greek mathematician Pythagoras, who is credited with its discovery. It has many practical applications in fields such as construction, engineering, and physics, and is also used in geometry and trigonometry.
Proof of the Pythagorean Theorem
There are several different proofs of the Pythagorean Theorem, but one of the most famous and elegant is the proof using similar triangles.
Consider a right triangle ABC, where C is the right angle and a, b, and c are the lengths of the sides opposite A, B, and C, respectively.
Now, draw a line segment from C to a point D on side AB, such that CD is perpendicular to AB.
This creates two smaller triangles, ACD and BCD, that are both similar to the original triangle ABC. This means that their corresponding angles are equal, and their corresponding side lengths are proportional.
Let x be the length of AD, and y be the length of BD. Then, we can write:
AC/AD = AB/BD
BC/BD = AB/AD
Since AC = a, BC = b, and AB = c, we can substitute these values and simplify:
a/x = c/y and b/y = c/x
Now, we can solve for x and y:
x = (ac) / sqrt(a^2 + b^2) and y = (bc) / sqrt(a^2 + b^2)
Finally, we can use the Pythagorean Theorem to find the length of CD:
CD^2 = AD^2 + BD^2
= x^2 + y^2
= (a^2c^2) / (a^2 + b^2) + (b^2c^2) / (a^2 + b^2)
= (a^2c^2 + b^2c^2) / (a^2 + b^2)
= (c^2)*(a^2 + b^2) / (a^2 + b^2)
Therefore, CD^2 = c^2, which means that CD = c. This proves the Pythagorean Theorem: a^2 + b^2 = c^2.
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Here’s an example of how to use the Pythagorean Theorem to find the length of one side of a right triangle.
Example: A right triangle has a hypotenuse of length 10 cm and one leg of length 6 cm. Find the length of the other leg.
Solution: Let’s call the length of the missing leg x. Then we can use the Pythagorean Theorem to set up an equation:
6^2 + x^2 = 10^2
Simplifying, we get:
36 + x^2 = 100
Subtracting 36 from both sides, we get:
x^2 = 64
Taking the square root of both sides, we get:
x = 8
Therefore, the length of the missing leg is 8 cm.
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What is the Pythagorean Theorem used for?
The Pythagorean Theorem is used to find the length of any side of a right triangle, as long as the lengths of the other two sides are known. It has many practical applications in fields such as construction, engineering, and physics, and is also used in geometry and trigonometry.
Who discovered the Pythagorean Theorem?
The Pythagorean Theorem is named after the ancient Greek mathematician Pythagoras, who is credited with its discovery.
Can the Pythagorean Theorem be used for non-right triangles?
No, the Pythagorean Theorem can only be used for right triangles. For non-right triangles, other trigonometric functions such as sine, cosine, and tangent must be used.
What is a Pythagorean triple?
A Pythagorean triple is a set of three positive integers that satisfy the Pythagorean Theorem. The most well-known Pythagorean triple is (3, 4, 5), which corresponds to a right triangle with sides of length 3, 4, and 5 units.
Can the Pythagorean Theorem be proved using algebra?
Yes, there are several algebraic proofs of the Pythagorean Theorem. These involve manipulating the algebraic expressions for the areas of the squares on the sides of the triangle.
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