Sir, is it possible that, In the first question, As A & B are the mid points and the line joining the midpoints in an Equilateral triangle must pass from the center of the triangle. If we consider this, then we get MN as Diameter of the circle.
Vijay, the line joining the midpoints will not pass through the centre – you are possibly confusing with the median! If we join all three midpoints we will get another smaller similar (i.e. equilateral) triangle – see this post, last part https://crackthecat.wordpress.com/2013/08/06/the-basic-proportionality-theorem/
Once we have proved it (which we have above), then in future we can directly use it because we know it will be applicable! Proving stuff to yourself is a key thing in Geometry – it helps you apply the right theorem/formula/concept at the right time.
SIR, there is a typo error in ques 1… it should be …. let AM be x, and not MN be x
By anand madhav on August 29, 2013 at 9:37 AM
Anand, I’ve corrected it now. Thanks for pointing out! 🙂
regards
J
By catcracker on August 29, 2013 at 1:58 PM
I think the answer to 2nd part of 2nd question should be 1+sqrt5/2 not minus. Please take a look.
Regards,
K 😉
By latticesam on September 12, 2013 at 11:34 PM
The ratio should be less than 1….(1+rt5)/2 gives the ratio in the reverse direction.
regards
J
By catcracker on September 12, 2013 at 11:38 PM
ok 🙂 , so we have to pick the roots of the quadratic carefully .
By latticesam on September 12, 2013 at 11:44 PM
Don’t we always? 🙂 Here’s another such case where we have to be cautious…one of my earliest posts. https://crackthecat.wordpress.com/2013/01/15/recursive-expressions-1/
regards
J
By catcracker on September 12, 2013 at 11:53 PM
Sir, is it possible that, In the first question, As A & B are the mid points and the line joining the midpoints in an Equilateral triangle must pass from the center of the triangle. If we consider this, then we get MN as Diameter of the circle.
By vijay on October 21, 2013 at 12:53 PM
Vijay, the line joining the midpoints will not pass through the centre – you are possibly confusing with the median! If we join all three midpoints we will get another smaller similar (i.e. equilateral) triangle – see this post, last part https://crackthecat.wordpress.com/2013/08/06/the-basic-proportionality-theorem/
regards
J
By catcracker on October 21, 2013 at 1:23 PM
Sir,
In the 2nd sum, with prior knowledge of golden ratio, we could have easily said that the answer would be the reciprocal of the golden ratio!
By Phenom on October 17, 2016 at 6:00 AM
Exactly! Which is why prior knowledge is sometimes such a useful thing… 🙂
regards
J
By catcracker on October 17, 2016 at 1:00 PM
Hi J,
How can we use the Golden Ratio directly for the 2nd question.
By Yash on May 26, 2017 at 9:10 PM
Once we have proved it (which we have above), then in future we can directly use it because we know it will be applicable! Proving stuff to yourself is a key thing in Geometry – it helps you apply the right theorem/formula/concept at the right time.
regards
J
By catcracker on May 29, 2017 at 1:41 PM