# Integer-sided isosceles triangle with area equal to $120$

BdMO National 2013 Junior Q. 2:

Two isosceles triangles are possible with 120 square unit area of each and length of edges are integers. Such one is with 17, 17 and 16 unit edges. Determine the length of edges of second one. [Hint: In $\bigtriangleup ABC$ if $AB = AC$ and $AD$ is perpendicular to $BC$ then $BD = CD$ .]

Please help me, or just give me a hint to solve this problem.

## 2 Answers

Take the solution you have, cut it along the axis of symmetry and rearrange to another isosceles triangle.

• Can you just explain it a little more? – Ruhan Habib Feb 8 '14 at 8:44
• Sketch $\triangle ABC$. As mentioned in the hint, the altitude $AD$ bisects the triangle into equal halves, in fact with integer sides. Rearrange these two pieces. Better to draw and figure out, verbal description is perhaps more complex than drawing. – Macavity Feb 8 '14 at 8:52
• Thanks. I think I've found the answer right now. I think the base would be 15 and the other edges would be 17. Is it correct? By the way, thanks. – Ruhan Habib Feb 8 '14 at 9:04
• I should improve my geometry. I'm so bad at it! Can you suggest some website for improving my geometry skills? – Ruhan Habib Feb 8 '14 at 10:17
• You may want to check artofproblemsolving.com/Wiki/index.php/Olympiad_books#Geometry – Macavity Feb 8 '14 at 10:51

Cut the triangle along the axis of symmetry and rotate it to get another one. So, the answer is $30,17,17$.