Show that the equilateral triangle has congruent angles?

Question

This Question is of Chapter "Straight Line" the diagram of this question shows the values of ABC I am confused abut the values of C(x,y) it should be (b,c) but it's written something else can someone explain me why and how these values are produced.

Answer 1

I'm confused by your title, since the question you pose there differs from the question you post. I'll answer the question in your post, not in your title.

The coordinates of $C$ are given as $$ C\left( \frac{a}2, \frac{a\sqrt{3}}2\right) .$$ The $x$-coordinate is obtained by observing that $C$ lies at the same distance from $A$ as from $B$, so that it must lie on the perpendicular bisector of the segment $AB$. This perpendicular bisector is exactly the line with equation $$ x = \frac{a}2 .$$ To find the $y$-coordinate of $C$, we observe that it is just the height of the triangle. The height of an equilateral triangle is always $\frac{\sqrt3}2$ times the length of its side, giving $$ y = \frac{a\sqrt{3}}2 .$$ To show that the height of an equilateral triangle is always $\frac{\sqrt3}2$ times the length of its side, use the Pythagorean theorem in the triangle $ADC$ where $D(\frac{a}2,0)$. The theorem says that $$ \text{height}^2 + \left(\frac{a}2\right)^2 = a^2 ,$$ leading to $$ \text{height} = \frac{a\sqrt{3}}2 .$$