Can You Actually Solve This Viral Geometry Puzzle? (II)

In this essay, I have yet another viral geometry puzzle for you. I came across this puzzle on the internet recently and found it engaging. Again, as is often the case with viral math puzzles, I could not narrow down the definitive source. I had seen the puzzle from multiple sources before I decided to engage.

This geometry puzzle features a right-angled isosceles triangle with two equal sides measuring 4 units each. Inside the isosceles triangle is an inscribed circle that touches all three sides of the triangle.

Given these conditions, your challenge is to work out the area of the inscribed circle. Do you think you can figure it out?

Spoiler Alert:

Beyond this section, I will be explicitly discussing the solution to this puzzle. So, if you wish to try to solve this puzzle out on your own, I suggest you do it now.

Once you are done with your attempt, you may return to this essay and continue reading to compare our respective approaches.

This essay is supported by Generatebg

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Setting Up the Solution to the Viral Geometry Puzzle

The very first step that I would like to take is to label the significant vertices and points as shown in the illustration below. This would make our lives significantly easier as we proceed. Note that I have labelled the circle’s centre as ‘o’.

So far, we know the following:

AB = BC = 4 units

Using the Pythagorean theorem, we could figure out the unknown hypotenuse AC as follows:

AC² = AB² + BC²

→ AC² = 4² + 4² = 16 + 16 = 32

→ AC = √32 = √(2*16)

→ AC =4√2 units (considering positive lengths only)

Now that we know that AC is 4√2 units long, we may start building a mathematical relation between the circle and the sides of the triangle.

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Applying the Tangent-to-Circle Theorem to the Viral Geometry Puzzle

Consider the two equal sides of the triangle: AB and BC. From the given conditions, we know that these two sides are at right angles to each other. We could try and take advantage of this fact.

Let us connect the centre of the circle ‘o’ to the circle’s contact points to AB (D) and BC (E) respectively using straight lines.

The tangent-to-circle theorem states the following:

The tangent to a circle is always perpendicular to the circle’s radius.

If you were perceptive enough, you would have realized that OD and OE are both of equal length (since they are both radii). Using the insights we have so far, we end up with the square ODBE as shown below. Let us assign the variable ‘r’ to the unknown radius of the cirle.

Since ODBE is a square, DB and DE are equal to ‘r’ as well. Consequently, AD = (4 — r) units = EC. We are now a little bit closer to solving the puzzle.

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Applying the Two-Tangent Theorem to the Viral Geometry Puzzle

The two-tangent theorem to a circle states the following:

When two different tangents are drawn to a circle from the same external point, the segments that touch the circle from the external point are equal in length.

When we apply this theorem to the puzzle setup we already have, we realise that AD must be equal to FA, and EC must be equal to CF. Consequently, we get the following result:

AD = FA = EC = CF = (4 — r) units

We are almost there. Let us now go for the home stretch.

Solving the Viral Geometry Puzzle

We now have everything we need to solve this geometry puzzle. Let us focus on the hypotenuse: CA. We know that CA = 4√2 units. At the same time, we know that CA = CF + FA. As a result, we get the following expression:

CA = CF + FA = 4√2 (units)

→ CA = (4 — r) + (4 — r) = 4√2

→ 8–2r = 4√2

Subtracting 8 from both sides:

→ -2r = 4√2–8

Multiplying by (-1) on both sides:

→ 2r = 8 — (4√2)

Dividing by 2 on both sides:

r = [4 — (2√2)] units

We now have the measure of the circle’s radius. From this point, we could just plug this into the well-known circle’s area formula to compute its area:

Area of the circle = πr² = π[4 — (2√2)]² = 4.3121 units² (rounded-off to first four digits)

There you go. The area of the inscribed circle is (approximately) 4.3121 units².

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Further reading that might interest you: Can You Actually Solve This Viral Geometry Puzzle (I) and How To Really Solve This Tricky Algebra Problem Series?

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