Math Is Fun Forum

sologuitar

Member

Registered: 2022-09-19

Posts: 467

Vertices of Equilateral Triangle

Verify that the points (0, 0), (a, 0), and (a/2, sqrt{3}a/2) are the vertices of an equilateral triangle. Then show that the midpoint of the three sides are the vertices of a second equilateral triangle.

Any hints?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,236

Re: Vertices of Equilateral Triangle

First part is easy either by calculating the lengths of sides or by using sines/cosines to get the angles.

Second part: Well you could get the coordinates of the midpoints and calculate the lengths of the smaller triangle but that's a lot of work. There's an easier way.

Let's say that an equilateral triangle has vertices A, B and C and that the midpoints are D, E and F with D opposite A, E opposite B and F opposite C

Show that triangles ABC and AEF are similar and hence EF is 0.5 BC.

Similarly show ED and DF are half the side opposite each. Hence EF = DE = DF so DEF is also equilateral.

As this is true for all equilaterals it is true for the one in the above question.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

sologuitar

Member

Registered: 2022-09-19

Posts: 467

Re: Vertices of Equilateral Triangle

First part is easy either by calculating the lengths of sides or by using sines/cosines to get the angles.

Second part: Well you could get the coordinates of the midpoints and calculate the lengths of the smaller triangle but that's a lot of work. There's an easier way.

Let's say that an equilateral triangle has vertices A, B and C and that the midpoints are D, E and F with D opposite A, E opposite B and F opposite C

Show that triangles ABC and AEF are similar and hence EF is 0.5 BC.

Similarly show ED and DF are half the side opposite each. Hence EF = DE = DF so DEF is also equilateral.

As this is true for all equilaterals it is true for the one in the above question.

Bob

Ok. I will work this out before Saturday. I have several personal errands to run today. By the way, Suzzane Somers from Three's Company and Step By Step died on Monday October 15 this year one day before her 77th birthday. A beautiful, smart, talented, business lady, actress she was for sure. Love Somers as Chrissy Snow on Three's Company with the late John Ritter and the beautiful Joyce DeWitt. You see, death is no respector of social status.