Math CentralQuandaries & Queries


Question from Karan, a student:

I have been given an isosceles triangle. the Top angle is what i have to find out and the two sides adjacent to it are both 4.9 cm. i have been told that the area of the triangle is 4 cm^2. i have no idea how to work this out, any ideas?

Hi Karan,

I drew an isosceles triangle and I will use it to help you get started.


$D$ is the midpoint of $BC$ and hence, since triangle $ABC$ is isosceles the triangles $ACD$ and $ADC$ are congruent and angle $ADC$ is a right angle. Suppose that $h = |AD|,$ the height of the triangle and $b = |BC|,$ the length of its base. You know the area of the triangle is 4 square centimeters and hence

\[\frac12 b \times h = 4 \mbox{ square centimeters.}\]

Pythagoras Theorem applied to triangle $ADC$ hives a second equation in $b$ and $h.$ Solve the first equation for $h$ and substitute into the second. Square both sides and simplify. This gives you a fourth degree equation in $b$ but if you look closely it is a quadratic in $b^{2}.$ Solve this quadratic.

Can you complete the problem now? Write back if you need more assistance,

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