Question

Write a statement of Pythagoras theorem and show that the 6, 8 and 10 are Pythagorean triples.

Answer 1

Hint:
Here, we will first write the basic Pythagoras theorem for the right angled triangle. Then we will find the sum of the square of the two small sides given and check whether it is equal to the square of the third side or not. Then by solving this we can say that whether the given three sides are the Pythagorean triplets or not.

Complete step by step solution:
Pythagoras theorem states that the square of the hypotenuse is equal to the sum of the square of the other two sides i.e. base and perpendicular of a right angles triangle.
$${\left(\mathrm{H}\mathrm{y}\mathrm{p}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{u}\mathrm{s}\mathrm{e}\right)}^2={\left(\mathrm{B}\mathrm{a}\mathrm{s}\mathrm{e}\right)}^2+{\left(\mathrm{P}\mathrm{e}\mathrm{r}\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}\right)}^2$$

Therefore, by Pythagoras Theorem we can write
$${\left(AB\right)}^2={\left(BC\right)}^2+{\left(AC\right)}^2$$
The given three sides is 6, 8 and 10.
Now we will apply this Pythagoras theorem to check whether the given three sides are the Pythagorean triplets or not. We know that the hypotenuse is the longest side of a triangle. Therefore we will find the sum of the square of small two sides given and check whether it is equal to the square of the third side. Therefore, we get
$$\Rightarrow 6^2+8^2={10}^2$$
Applying the exponent on terms, we get
$$\Rightarrow 36+64=100$$
Adding the terms on left side, we get
$$\Rightarrow 100=100$$
We can clearly see that it satisfies the Pythagoras theorem.

Hence, 6, 8 and 10 are the Pythagorean triplets.

Note:
Here, we have to note that the Pythagoras theorem is only applied for the right angles triangle. Right angled triangle is the triangle which has one of its angles equal to the $${90}^{\circ }$$. Pythagoras theorem is generally used to find the third side of the triangle if any two sides of the triangle are given. Also trigonometric functions are used for the right angled triangle to find the angles of the triangle or the sides of the triangle.