Problem about equilateral triangles

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In summary, the ratio of the perimeter of the larger equilateral triangle to that of the smaller is 2√3/3, which can be found by using the fact that the triangles are similar and finding the ratio of their perimeters, which is equal to the ratio of their heights. By using the Pythagorean theorem, it can be shown that the altitude of the equilateral triangle is equal to s√3/2, where s represents the length of a side. This leads to the conclusion that the ratio of the perimeters is 2√3/3.
  • #1
Nich6ls
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Hi! I need help with this exercise: A side of one equilateral triangle equals the height of a second equilateral triangle. Find the ratio of the perimeter of the larger triangle to that of the smaller. A "detailed solution" to be analysed by me then.

Answer: \[ 2/3 sqrt3 \]

Thanks
 
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  • #2
Use the fact that triangles are similar. If $x$ represents the side of first triangle, show that the triangle's height is $x\frac{\sqrt{3}}{2}$. The second triangle has height $x$, and since the triangles are similar, the ratio of their perimeters is equal to the ratio of their heights. Conclude.
 
  • #3
If you drop a perpendicular from one vertex of an equilateral triangle to the opposite side, you divide the equilateral triangle into two right triangles that have hypotenuse equal to the length of a side of the equilateral triangle, one leg half that length, and the second leg, whose length you can get from the Pythagorean theorem, is the altitude of the equilateral triangle. If the length of a side of the original equilateral triangle is "s" then the altitude is $\sqrt{s^2- (s/2)^2}= \sqrt{s^2- s^2/4}= \sqrt{3s^2/4}= \frac{s\sqrt{3}}{2}$.

So the first equilateral triangle has perimeter 3s while the second has perimeter $\frac{3s\sqrt{3}}{2}$. The ratio of those is $\frac{3s}{\frac{3s\sqrt{3}}{2}}= 3s\frac{2}{3s\sqrt{3}}= \frac{2}{\sqrt{3}}= \frac{2\sqrt{3}}{3}$.

(What you wrote, "2/3sqrt(3)" would correctly be interpreted as $\frac{2}{3\sqrt{3}}$, which is wrong, but I suspect you meant "(2/3)sqrt(3)"or $\frac{2}{3}\sqrt{3}= \frac{2\sqrt{3}}{3}$, which is correct.)
 

Related to Problem about equilateral triangles

1. What is an equilateral triangle?

An equilateral triangle is a type of triangle with three equal sides and three equal angles. This means that all three sides are the same length, and all three angles are the same measure of 60 degrees.

2. What is the formula for finding the area of an equilateral triangle?

The formula for finding the area of an equilateral triangle is A = (s^2 * √3)/4, where s is the length of one side. This formula is derived from splitting the equilateral triangle into two congruent right triangles and using the Pythagorean theorem.

3. How many lines of symmetry does an equilateral triangle have?

An equilateral triangle has three lines of symmetry. This means that it can be rotated 120 degrees around its center and still look the same.

4. Can an equilateral triangle also be an isosceles triangle?

Yes, an equilateral triangle is a special case of an isosceles triangle. This is because it has two equal sides, which also means that it has two equal angles.

5. What are some real-life examples of equilateral triangles?

Some real-life examples of equilateral triangles include the shape of a traffic sign, the pyramids in Egypt, and the faces of a regular octahedron. It is also commonly seen in architecture, art, and nature.

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