top button
Flag Notify
    Connect to us
      Site Registration

Site Registration

Two triangles have integral side lengths, with all sides being less than 50. They are similar but not congruent...

0 votes
216 views

Two triangles have integral side lengths, with all sides being less than 50. They are similar but not congruent and smaller triangle has two side lengths identical with the larger triangle.

What is the sum of the side lengths of the larger triangle?

posted May 11, 2016 by Rajni

Share this puzzle
Facebook Share Button Twitter Share Button LinkedIn Share Button

1 Answer

0 votes

the answer is 57
and logic is -
Let the smaller triangle have side lengths a, b, c with a < b < c, and the larger triangle have side lengths b, c, d with b < c < d.

a, b, c, d form a geometric progression, and (a +b) > c. So the ratio is less than 1.618. Since all the sides are integral, a = f^3, b = f^3r (where r is the ratio), c = f^3r^2, and d = f^3r^3.

Since d < 50, d is an integer cube > 2, d must be 27. The ratio r therefore must b 1.5 so that a = 8: b = 12, c = 18.

12 + 18 + 27 = 57

answer May 13, 2016 by Hasan Raza



Similar Puzzles
0 votes

A square with side length 1 is divided into 4 congruent right triangles, as shown, and a square in the center. Inscribe a circle in each triangle and in the center square. If all 5 circles are congruent, what is the radius of each circle?
enter image description here

–1 vote

A given parallelogram has sides measuring 7 and 9, and both its diagonals have integer lengths. Find the sum of all possible products of the lengths of the diagonals.

...