Question

The sides of a triangle are in the ratio $3:4:5$ then the measure of the greatest angle is

Answer 1

Hint: From the question given we have to find the measure of the greatest angle whose sides are in the ratio $3:4:5$. By observing the ratio of the sides, we can conclude that the triangle is the right angle because the ratio of the sides is a Pythagorean triplet. In right angle triangle one angle is ${{90}^{\circ }}$ and the sum of the remaining two sides is equal to the ${{90}^{\circ }}$ . so, the measure of greatest angle is ${{90}^{\circ }}$.

Complete step by step solution:
From the question given that the sides of a triangle is
$\Rightarrow 3:4:5$

Let the sides of a triangle is
$\Rightarrow 3x,4x,5x$
By observing the sides, we can conclude that the sides are Pythagorean triplet.
As we know that the Pythagorean triplet means it consists of three positive integers a, b, and c, such that
$\Rightarrow {{a}^{2}}+{{b}^{2}}={{c}^{2}}$
Here, the given sides are also in this form only,
$\Rightarrow {{\left( 3x \right)}^{2}}+{{\left( 4x \right)}^{2}}={{\left( 5x \right)}^{2}}$
By simplifying further, we will get,
$\Rightarrow 9{{x}^{2}}+16{{x}^{2}}=25{{x}^{2}}$
 By simplifying further, we will get,
$\Rightarrow 25{{x}^{2}}=25{{x}^{2}}$
Therefore, hence proved the given ratio of sides are Pythagorean triplet.
By this we can say that the triangle is right angle triangle, as we know that in right angle triangle one angle is ${{90}^{\circ }}$ and the sum of the remaining two sides is equal to the ${{90}^{\circ }}$ . so, the measure of greatest angle is,
$\Rightarrow {{90}^{\circ }}$

Note: Students should know about the Pythagorean triple, and student should also remember that the Pythagorean triplets will always form a right angled triangle, and student should also know that the greatest angle will always be opposite to the greatest side.