Can You Actually Solve This Viral Geometry Puzzle?

I recently came across this viral geometry puzzle on the internet, and found it interesting to solve. I tried to track down the original source of this puzzle. But as is often the case with viral content, it is very difficult to narrow down the source. I personally came across this puzzle first from the work done by Presh Twalwalkar, who in turn attributes it to someone named Reio from Romania.

What you have in this geometry puzzle is a square with an arbitrary-looking point inside it. This point in turn connects to the mid-point of each of the square’s sides via line-segments. This splits the square into four uneven areas.

The top-left area amounts to 10 cm², whereas the top-right area amounts to 16 cm² and the bottom-left area to 8 cm². Given these conditions, your challenge is to figure out the area of the remaining bottom-right region. Do you think that you can work this out?

Spoiler Alert:

If you wish to give this puzzle a try on your own, I suggest that you pause reading this essay now. Once you are done with your attempt, you may return and continue reading.

Beyond this point, I will be explicitly discussing the solution to this puzzle.

This essay is supported by Generatebg

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The Key to Solving this Viral Geometry Puzzle

We start with the well-known formula for calculating the area of an arbitrary triangle using its base and height:

Area of a triangle = ½ * (Base) * (Height)

Following this, we extend this formula to assert that triangles of different shapes that share the same base and height will occupy the same area.

Consider the illustration below, where you have three differently shaped triangles that share the same base and height. They would each yield the same area as the other.

With this insight revealed, we are now ready to attack this puzzle.

Triangles inside the Square

We have so far been discussing triangles. But the given puzzle features a square and four quadrilaterals. So, we need to construct some triangles in there. To do this, we could just connect the intersecting point to all four corners of the square. This would result in 8 triangular regions inside the square as you can see below:

Now, consider the two bottom triangles shaded in blue in the illustration below. These two share the same base (the bottom side of the square) and the same height (limited by the intersecting point).

Using our insight from earlier, we could assert that these two triangles would have the same area (which I have marked as ‘a1’ for each). Via the same logic, we would end up with three more pairs of triangles with equal areas (‘a2’, ‘a3’, and ‘a4’ respectively).

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How to Really Solve this Viral Geometry Puzzle

Having established the area for each triangle, I choose to sum up the areas of triangles that lie along the left-diagonal of the square (bottom to top) as follows:

Sum of areas along the left-diagonal = (a1 + a4) + (a2 + a3)

Next, let us see what happens when we sum up the areas of the triangles that lie along the right-diagonal of the square (bottom to top):

Sum of areas along the right-diagonal = (a1 + a2) + (a3 + a4)

So, it turns out the sum of the areas along the opposite diagonals are equal to each other:

Sum of areas along the left-diagonal = Sum of areas along the right-diagonal

→ (a1 + a4) + (a2 + a3) = (a1 + a2) + (a3 + a4) →A

Let us hold this equation (A) at the back of our minds for now. From the information given to us in the puzzle, we know the following:

a1 + a4 = 8 cm²

a3 + a4 = 10 cm²

a2 + a3 = 16 cm²

a1 + a2 = ??

When we plug in these values into equation A, we get the following result:

(a1 + a4) + (a2 + a3) = (a1 + a2) + (a3 + a4)

(8 + 16) cm² = (a1 + a2) + 10 cm²

Subtracting 10 cm² from both sides:

a1 + a2 = 14 cm²

There we go. The area of the bottom-right region of the square is 14 cm² and the total area of the square is 48 cm².

We started with the fact that triangles of different shapes that share the same base and height occupy the same area. We then used this insight to established that the sum of the areas along the opposite diagonals are equal. The resulting equation enabled us to sovle for the unknown area.

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Further reading that might interest you: How To Really Benefit From Curves Of Constant Width? and What Really Happens When You Divide By Zero?

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