Geometry Puzzle – Challenge 76

Challenge:

If the perimeter of an equilateral triangle is 2x meters and its area is x square meters, then what is the length of one side of the triangle in meters?

A- \(\sqrt{3}\)

B- \(\frac{\sqrt{3}}{2}\)

C- \(2\sqrt{3}\)

D- \(\frac{2\sqrt{3}}{3}\)

E- 3

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The perimeter of the equilateral triangle is 2x meters. So, one side is \(\frac{2}{3}x \) meters.
The area of an equilateral triangle \(= \frac{s^2 \sqrt{3}}{4}\) (s is one side of the triangle)
The perimeter of the triangle is twice its area. So:
\(2x = 2 (\frac{s^2 \sqrt{3}}{4}) → 2x = (\frac{s^2 \sqrt{3}}{2})\)
Replace the s with \(\frac{2}{3}x\). Then:
\(2x = \frac{(\frac{2}{3} x)^2 \sqrt{3}}{2} = \frac{\frac{4}{9} x^2 \sqrt{3}}{2 }→ 4x = \frac{4}{9} x^2 \sqrt{3} → 4 = \frac{4}{9} x\sqrt{3} → 9 = x\sqrt{3}→
\frac{9}{\sqrt{3} }= x → \frac{9}{\sqrt{3} } × \frac{\sqrt{3}}{\sqrt{3} } = x → x = 3\sqrt{3}\)
Then, one side of the triangle is: \(\frac{2}{3}x =\frac{ 2}{3}(3\sqrt{3}) = 2\sqrt{3}\)

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Frequently Asked Questions

How do I find one side of an equilateral triangle from its perimeter?

Divide by 3. For perimeter 2x: side = 2x/3.

What is the area formula for an equilateral triangle?

A = (s^2 sqrt(3))/4, where s is the side length. The formula follows from splitting the triangle into two 30-60-90 right triangles.

How do I set up the equation?

Substitute s = 2x/3 into A = (s^2 sqrt(3))/4: A = ((2x/3)^2 sqrt(3))/4 = (4x^2/9)(sqrt(3)/4) = x^2 sqrt(3)/9. Set equal to the given area: x^2 sqrt(3)/9 = x.

How do I solve x^2 sqrt(3)/9 = x?

Divide both sides by x (valid since x > 0): x sqrt(3)/9 = 1, so x = 9/sqrt(3) = 9 sqrt(3)/3 = 3 sqrt(3).

How do I rationalize 9/sqrt(3)?

Multiply numerator and denominator by sqrt(3): 9/sqrt(3) = 9 sqrt(3)/3 = 3 sqrt(3).

What is the side length?

Side = 2x/3 = 2(3 sqrt(3))/3 = 2 sqrt(3) meters.

How do I check the answer?

With side 2 sqrt(3): perimeter = 3 times 2 sqrt(3) = 6 sqrt(3). Area = ((2 sqrt(3))^2 sqrt(3))/4 = (12 sqrt(3))/4 = 3 sqrt(3). Check: perimeter 6 sqrt(3) = 2 times area 3 sqrt(3) ✓.

Why is the equilateral area formula different from a general triangle?

Because for an equilateral triangle the height can be computed exactly from the side using the 30-60-90 ratio: h = s sqrt(3)/2. Plug into A = (1/2)(s)(h) to get s^2 sqrt(3)/4.

What if perimeter were 3x and area x?

Side = 3x/3 = x. Area = x^2 sqrt(3)/4 = x, so x sqrt(3)/4 = 1, x = 4/sqrt(3) = 4 sqrt(3)/3.

Where does the equilateral triangle area formula come from?

From splitting the triangle into two 30-60-90 right triangles. The 30-60-90 side ratios (1 : sqrt(3) : 2) give the exact height as (sqrt(3)/2) times the side.

Related Lessons You May Like

• How to find the area of rectangles
• How to find the perimeter of rectangles
• How to find the area of triangles
• How to solve systems of equations
• How to solve multi-step word problems

If your student enjoys these puzzles, Geometry for Beginners works the same relationships inside a full curriculum. Pre-Algebra for Beginners covers the algebra foundations.