Part A This is the start of an explanation of how to plan a coordinate proof that all equilateral triangles are similar. Choose the vertices necessary. Show that an equilateral triangle with side length a can be placed on the coordinate plane with these vertices. A. $$ (0,0) ;(0, a) ;\left(-\frac{a \sqrt{3}}{2}, 0\right) $$ B. $$ (-a, 0) ;\left(-\frac{a}{2}, 0\right) ;\left(0,-\frac{a}{2}\right) $$ C. $$ \left(-\frac{a}{2}, 0\right) ;\left(\frac{a}{2}, 0\right) ;\left(0, \frac{a \sqrt{3}}{2}\right) $$ D. $$ (a, 0) ;\left(\frac{a \sqrt{3}}{2}, 0\right),\left(0, \frac{a}{2}\right) $$ Part B Choose the words to finish the explanation of how to plan a coordinate proof that all equilateral triangles are similar. Show that an equilateral triangle with side length a can be placed on the coordinate plane with the vertices specified above. Similarly place an equilateral triangle with legs having length $$ b $$ on the coordinate plane. Show that there exists a $$ \square $$ that maps one triangle onto the other. Choose... reflection dilation Roviow Progress rotation translation Question 7 of 17 Back Next

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UpStudy AI · Accepted Answer

For Part A, you need to choose vertices that define an equilateral triangle with side length a. Option C, with vertices (–a/2, 0); (a/2, 0); (0, (a√3)/2), is the correct choice. For example, the distance between (–a/2, 0) and (a/2, 0) is

√[(a/2 – (–a/2))² + (0 – 0)²] = √[(a)²] = a,

and the other side lengths can be verified similarly.

For Part B, after placing one equilateral triangle with side length a and a second one with side length b on the coordinate plane, you show that a dilation is the transformation that maps one triangle onto the other (preserving the angles and scaling the side lengths by a constant factor).

Thus, the correct answers are:
Part A: Option C  
Part B: dilation
For Part A, choose option C: (–a/2, 0), (a/2, 0), and (0, (a√3)/2) as the vertices of the equilateral triangle. For Part B, use dilation to map one equilateral triangle onto the other.

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