Question

The sides of a triangle are in the ratio 4:6 & :7. Then

A.The triangle is obtuse-angled

B.The triangle is acute-angled

C.The triangle is right-angled

D.The triangle is impossible

Answer 1

Hint: Draw the diagram with the help of sides ratios given and let the sides be 4x,6 & x,7x then we check below conditions,
If the square of sum of two sides is greater than the square of the third side, there will be an acute angle triangle.
If the square of the sum of two sides is the same as the square of the third side, there will be a right angle triangle.
If the square of the sum of two sides is smaller than the square of the third side, there will be an obtuse angle triangle.

Complete step-by-step answer:
As per the given sides ratios are 4:6 & :7
Diagram:

Hence, let the common parameter for sides be x
So, the sides can now be given as, 4x,6 & x,7x
Hence, now calculate the square of the sum of two smaller sides and comparing it with the square of the third side we can determine the triangle.
The square sum of two smaller side is
\[ = {\left( {4x} \right)^2} + {\left( {6x} \right)^2}\]

On opening the brackets, we get,
\[ = 16{x^2} + 36{x^2}\]
On addition, it is \[52{x^2}\]
While the square of third side is
\[ = {\left( {7x} \right)^2}\]
Which can be simplified as to \[49{x^2}\]
Hence, the square of sum of two sides is greater than square of third side as \[52{x^2} > 49{x^2}\]
So, there will be an acute angle triangle.
Hence, option (B) is the correct answer.

Note: Triangles can also be classified by their angles. In an acute triangle all three angles are acute (less than \[90\]degrees). A right triangle contains one right angle and two acute angles. And an obtuse triangle contains one obtuse angle (greater than \[90\]degrees) and two acute angles.
Classifying a triangle is as simple as comparing the sides. If all three sides have the same length then it is an equilateral triangle, if only two sides have the same length then it is an isosceles triangle and if there are no sides that have the same length then it is a scalene triangle.