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Thursday, June 18, 2015

UPSC Maths Problems Challenge : A Problem A Day! (Day 7)

This is a very interesting problem and a  problem which can be categorised under both Geometry and Permutation And Combination.I have provided a detailed solution to this,which is a rare case online since the globalised generation does not really give too much of a damn about providing detailed solutions to questions at most of the websites.I hope this comes of use to all those nerds who are working day and night on their huge Maths syllabus topped with UPSC preparation.

Today's Problem(19/06/2015)

Question Credit :

Which of the following triangles with measures of sides given below is an obtuse angled triangle?
a)12, 5, 13
b)6, 7, 5
c)7, 3, 5
d)8, 8, 8

Let us first learn how you can find out if a triangle is acute angled,right angled or obtuse angled. 
For checking if it is a right angled triangle or not is easy,we just need to use the Pythagorean Theorem,i.e (perpendicular)^2 + (base)^2 = (hypotenuse)^2,though it is certainly better to remember the first few Pythagorean triplets.
Next is the big deal.How do you find out if it is an acute angled triangle or obtuse angled triangle?
You just simply add the squares  of the shortest sides of the triangle and find the square root of the resultant number.If you find it is greater than the longest side provided,it will be an acute angled triangle and if you find it is shorter than the longest side provided,it will be an obtuse angled triangle.

In this problem,Option a) clearly speaks of a right angled triangle,since 5,12 and 13 is a Pythagorean Triplet.
Option d) clearly speaks of an equilateral triangle as all the sides are equal.So,each angle is equal to 60°.
The square of sum of the shortest sides of option b) combination is 61.
And root over 61 = 7.81024,which is more than 7.You can apply the same logic for option c) but the square root of the resultant number,i.e root over 34 is less than 7,the longest side provided.Therefore,option c) speaks of an obtuse angled triangle.
So,option C is correct.

Here is a relevant article on Brilliant how you can square large numbers easily,without brainstorming too much.Do give it a read.
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