Two congruent right triangles share a vertex. If the overlap represents 1/7 of the total area, what’s the angle?

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## Solution

The angle α=30°.

Because of angle comparison we find that the overlap triangle is similar to the two large ones. We can then place the lengths as shown.

Now the overlap area is sin^{2}(α)tan(α)/2. The total area is two times the area tan(α)/2 minus the overlap. The fraction is easily worked out as sin^{2}(α)/(2-sin^{2}(α)).

Setting this fraction to 1/7, we find sin^{2}(α)=1/4 and since α is acute we find sin(α)=1/2 and hence α=30°.

## Poem

The horse’s head is inspiring

The horse whisperer is talking

To his best friend

A mustang strong and loving

The horse is understanding.

Everyday early in the morning

The whisperer and the horse are practising

And that’s why love between horse and man is beginning