Question

prove that area of equilateral triangle in root 3 upon 2 a
square unit

Answer 1

In an equilateral triangle, the height will cut across the middle of the triangle, forming two right triangles and dividing the base in 2 equal halves.


Proof:
Step 1: Since all the 3 sides of the triangle are same,

AB = BC = CA = a

Step 2: Find the altitude of the △△ABC. 

Draw a perpendicular from point A to base BC, AD ⊥⊥ BC

By using Pythagoras theorem

In △△ ADC

h22 = AC22 - DC22

= a2a2 - (a2)2(a2)2 [Because, DC = a2a2 ]

= a2a2 - a24a24

h = 3√a23a2

Step 3: We know that, Area of a triangle = 1212 * Base * Height

= 1212 * a * 3√a23a2

= 3√434a2a2

The area of a equilateral triangle = 3√434a2a2.

Answer 2

 In ΔABD and ACD, we have

AB = AC

∠ADB = ∠ADC  [ΔABC is equilateral]
 (Each equal to 90 )
AD = AD

So, by RHS criterion of congruence,

ΔABD ΔACD

⇒ BD = DC

But, BD + DC = a

⇒ BD = DC =