Question

can someone please help? the question is an equilateral triangle is sometimes/always/never similar to a scalene triangle, because we can sometimes/always/never map one onto the other using only dial actions rigid transformations.

Answer 1

Answer:

An equilateral triangle is never similar to a scalene triangle, because we can never map one onto the other using only dilations and rigid transformations.

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Explanation:

An equilateral triangle is one where it has all three sides the same length. "Equi" means "equal" while "lateral" means "side". Any equilateral triangle also has all three angles the same measure (we consider it equiangular). Each angle is 60 degrees.

In contrast, a scalene triangle has all of its sides different, so consequently that means all of its three angles must be different as well. If you had exactly two angles congruent to one another, then the opposite sides would be a congruent pair of sides, but then that leads to an isosceles triangle. Also, if all three angles were the same, then we'd get back to an equilateral triangle. So in short, any scalene triangle has all of its angles different.

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Let's consider two cases

• Case 1) The scalene triangle has one angle that is 60 degrees, while the other two angles are not 60 degrees.
• Case 2) The scalene triangle doesn't have any angles that are 60 degrees.

If case 1 happens, then we could match up one pair of 60 degree corresponding angles between the equilateral triangle and the scalene triangle. However, we won't have any more matches. This is because the other two angles aren't 60 (if they were then the triangle isn't scalene). Therefore case 1 leads to dissimilar triangles.

If case 2 happens, then we get the same conclusion as case 1. This time we don't have any angles matching up. Similar triangles must have corresponding angles match up and be congruent. Put another way: similar triangles are the same shape but not necessarily the same size.

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After completing both cases, we can see that it is impossible to have an equilateral triangle be similar to a scalene triangle. In terms of transformations, we cannot apply any of the following

• Translation
• Reflection
• Rotation
• Dilation

and have one triangle turn into the other. Again, it's all because the angles do not match up. We may have one matching pair, but we need all three pairs to match.