ISI2018-MMA-1

Question

The number of isosceles (but not equilateral) triangles with integer sides and no side exceeding $10$ is

• A. $65$
• B. $75$
• C. $81$
• D. $90$

Answer 1

In a triangle sum of any two sides is greater than the third side.

We can use brute force here, fix one side and make other two sides equal.

fixing one side = 1, other two sides can be any of the nine possible numbers => 9 ways

fixing one side = 2, other two sides have 8 options

similarily, for 3, 8 ways

for 4, 7 ways

for 5, 7 ways

for 6, 6 ways

for 7, 6 ways

for 8, 5 ways

for 9, 5 ways

for 10, 4 ways

total ways = 65

Answer 2

 no. of isosceles triangles with integer sides, no sides exceeding n= 1/4(3n^2+1) or 3/4(n^2) according as n is odd or even, n is any integer.

now n=10 even

no. of isosceles triangles=3/4*(10^2)=3*25=75

Answer=B

 

Comments

Answer will be 65. Because it is mentioned that (but not equilateral)

So, you need to remove the 10 equilateral triangles also.