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The diagram shows an equilateral triangle $ADE$ inside a square $ABCD$ . What is the value of $frac{[Delta ADE]}{[Delta DEC]}$ .

Mathematics Asked on February 17, 2021

The diagram shows an equilateral triangle $ADE$ inside a square $ABCD$ . What is the value of $frac{[Delta ADE]}{[Delta DEC]}$ .

What I Tried: Here is the diagram :-

You can see I marked the angles as required. Now let $AB = x$ . We then have :-
$$[Delta ADE] = frac{sqrt{3}}{4}x^2$$
Now from here :- https://www.quora.com/What-is-the-ratio-of-sides-of-a-30-75-75-angle-triangle-without-sine-rule , I could understand and show that :- $$ED : DC : CE = bigg(frac{sqrt{3} + 1}{2} : frac{sqrt{3} + 1}{2} : 1bigg)$$

So let $EC = k$ , $CD = DE = frac{(sqrt{3} + 1)k}{2}$ .

From here :- $$x = frac{(sqrt{3} + 1)k}{2}$$
$$rightarrow k = EC = frac{2x}{(sqrt{3} + 1)}$$

Now, we can find area by Heron’s Formula. We have :- $$s = x + frac{x}{(sqrt{3} + 1)}$$
$$rightarrow s = frac{xsqrt{3} + 2x}{(sqrt{3} + 1)}$$
So :- $[Delta DEC] = sqrt{s(s-a)(s-b)(s-c)}$

$$rightarrow sqrt{Bigg(frac{(xsqrt{3} + 2x)}{(sqrt{3} + 1)}Bigg)Bigg(frac{x}{(sqrt{3} + 1)}Bigg)Bigg(frac{x}{(sqrt{3} + 1)}Bigg)Bigg(frac{(xsqrt{3})}{(sqrt{3} + 1)}Bigg)}$$

This looks like really a complicated expression, and I really am not going to attempt to simplify this. So can anyone give me a different solution?

Thank You.

3 Answers

If $a,b$ are sides of a triangle and $x$ the mesure of angle between them, then the area of it is $${acdot b cdot sin xover 2}$$

We use that formula here.

We have $AD = AE =DE =DC=a $ so$$frac{[Delta ADE]}{[Delta DEC]} = {{a^2sin 60 over 2}over {a^2 sin 30 over 2}} = sqrt{3}$$

Correct answer by Aqua on February 17, 2021

$DC=x$ and the altitude of $Delta DEC$ from $E$ is $frac{x}{2}$. This gives us $$[Delta DEC] = frac{x^2}{4}$$ And the result $$frac{[Delta ADE]}{[Delta DEC]}=sqrt{3}$$

Answered by Daniel Mathias on February 17, 2021

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By Area of triangle using trigonometry

Area (DEC) $ = frac12 (CD)(DE) sin (30^circ) = frac 14 x^2$

Answered by Ak. on February 17, 2021

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