Question

The angles of a triangle are in the ratio 1:3:5 then the greatest angle is [Kerala (Engg.) 2002]
A. $\frac{5\pi }{9}$
B. $\frac{2\pi }{9}$
C.$\frac{7\pi }{9}$
D. $\frac{11\pi }{9}$

Answer 1

Hint:
In this question, we have given the angles of a triangle in the ratio 1:3:5. To find the greatest angle consider the given ratios. Simplify the ratios by multiplying the provided ratio by an unknown variable. Secondly, use the fact that a triangle's total angles equal 180 to form a linear equation. Lastly, compare the obtained result with the option provided.

Formula Used:
Sum of angles of triangles = $180^{o}$

Complete step-by-step solution:
Let x,3x, and 5x be the angles of the given triangle, then according to the property of the triangle we can write; x+3x+5x=${{180}^{o}}$
$9x={{180}^{o}}$
$x={{20}^{o}}=\frac{\pi }{9}$
Substituting the value of x we get;
$x=\frac{\pi }{9}$
$3x=3 \times \frac{\pi }{9}=\frac{3\pi }{9}$
$5x=5 \times \frac{\pi }{9}=\frac{5\pi }{9}$
Hence, the greatest angle is $\frac{5\pi }{9}$.

So, option A is correct.

Note:
In such a question student usually makes silly mistakes in computing the ratios. Use a common variable to solve the ratio. We multiplied the provided ratio by an unknown variable in order to solve this problem. Moreover, students should always keep in mind that a triangle is right-angled if one of the angles is 90 degrees and acute if the other angles are all less than 90 degrees.

Additional Information:
One of the most commonly applied properties in geometry is the triangle's angle sum property. Most often, the unknown angles are calculated using this attribute. The angle sum property of a triangle states that the sum of a triangle's three internal angles is 180 degrees. A closed shape with both interior and exterior angles, a triangle is made up of three line segments. When the values of the other two angles are known, one can apply the angle sum property to determine the measure of an unknown interior angle.