Answer:
BD = 10√3 units
AC = 20 units
Area ΔABC = 100√3 units²
Step-by-step explanation:
Interior angles of a triangle sum to 180°:
[tex]\implies \angle A + \angle B + \angle C =180^{\circ}[/tex]
[tex]\implies 60^{\circ} + \angle B +60^{\circ} =180^{\circ}[/tex]
[tex]\implies \angle B +120^{\circ} =180^{\circ}[/tex]
[tex]\implies \angle B +120^{\circ} -120^{\circ}=180^{\circ}-120^{\circ}[/tex]
[tex]\implies \angle B =60^{\circ}[/tex]
As the interior angles of an equilateral triangle are congruent, and ∠A=∠B=∠C then ΔABC is an equilateral triangle.
In an equilateral triangle, all three sides have the same length.
[tex]\implies AC=BC=AB=20\; \sf units[/tex]
Height of an equilateral triangle:
[tex]\boxed{\textsf{h}=\dfrac{\sqrt{3}}{2}a}[/tex]
Where a is the side length of the triangle.
As BD is perpendicular to AC, BD is the height of the triangle.
[tex]\begin{aligned}\implies BD & = \dfrac{\sqrt{3}}{2}(20)\\ & =10\sqrt{3}\; \sf units \end{aligned}[/tex]
Area of an equilateral triangle:
[tex]\boxed{\textsf{A}=\dfrac{\sqrt{3}}{4}a^2}[/tex]
Where a is the side length of the triangle.
Therefore:
[tex]\begin{aligned}\implies \textsf{Area of $\triangle ABC$} & =\dfrac{\sqrt{3}}{4}(20)^2\\& = \dfrac{\sqrt{3}}{4}(400)\\& = 100\sqrt{3}\; \sf units^2\end{aligned}[/tex]
