SOLUTION: Given square ABCD, let P and Q be the points outside the square that make triangles CDP and BCQ equilateral. Prove that triangle AP Q is also equilateral.

Algebra ->  Polygons -> SOLUTION: Given square ABCD, let P and Q be the points outside the square that make triangles CDP and BCQ equilateral. Prove that triangle AP Q is also equilateral.      Log On


   

Question 1201144: Given square ABCD, let P and Q be the points outside the square that make triangles
CDP and BCQ equilateral. Prove that triangle AP Q is also equilateral.
Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

Start with square ABCD.


Form equilateral triangles CDP and BCQ, such that the new points are outside the square.


Then form triangle APQ shown in red.

The goal is to prove that triangle APQ is equilateral.

For now let's focus on triangle ABQ.
BQ = BC (since BCD is equilateral)
AB = BC (since ABCD is a square)
AB = BQ (by the transitive property)

The AB = BQ statement then tells us triangle ABQ is isosceles.
The vertex angle is 90+60 = 150 degrees.
The base angles are (180-150)/2 = 30/2 = 15 degrees each.

Through similar logic, you'll find that triangles ADP and PCQ are congruent to isosceles triangle ABQ.
All isosceles triangles mentioned have the same base length
AQ = AP = PQ

Therefore, triangle APQ is equilateral.