A triangle has sides measuring 13, 13 and 10 unites. A second triangle is drawn using sides with 13, 13, and x units. If both triangles have the same area but they are not congruent, what is x equal to?


Thank you!

 May 17, 2021

using herons formula we get that the area of the first triangle is 60. We then plug this into heron's formula again, except with the third side as x this time, and the area. from this we get x = x = +/- 10 and x= +/- 24. X is 24.

 May 17, 2021

thank you very much. Appreciate it but what is herons formula?

 May 17, 2021

It is a formula to find the area of a triangle, given the sides. Google has a good explanation of exactly what it is.

BWStar  May 18, 2021

Thanks BWStar

You can use heron's formula but think about what I am saying anyway :)


The only way this can happen is if one triangle is acute and the other is obtuse.


there are many ways to tackle this problem, most are quite difficult but the answer can also be found using only basic logic and Pythagoras's theorem.


Draw the triangle 13,13, 10,    (Yes, you MUST draw it)

Cut it down the middle to form 2 congruent right-angled trianges.

For each of these right-angled triangles the hypotenuse is 13 and the base is 5, using pythag, the height is 12


I need another right-angled triangle with hypotenuse 13 and with the same area


How about if the base is 12 and the height is 5.  That has to work    (draw it to see what I mean)


So the 3rd side of the second triangle (before it is cut in half) must be 12+12 = 24

 May 18, 2021
edited by Melody  May 18, 2021

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