Answer:

Proved , AD = a√3

area of ABC = a²√3​

Step-by-step explanation:

In Triangle ABC, AB=BC=CA = 2a

That means AB=BC=CA and ABC is an equilateral triangle
Now AD is perpendicular to BC
So angle ADB = 90 DEGREE AND angle ADC = 90 Degree

Now In triangle ADB , triangle ADC,

AB=CA

angle ADB=angle ADC(=90 degree each)

AD = AD(Common)

Therefore , Triangle ADB is congruent to Triangle ADC(RHS rule)
So , BD = CD(CPCT)

BD + CD = BC

BD+BD = BC

2BD = BC

2BD=2a (BC = 2a)

BD= a

BD = CD = a

Now

i) In triangle ADB

AB^2 = AD^2+BD^2

(2a)^2 = AD^2 + a^2

4a^2 - a^2 = AD^2

3a^2 = AD^2
AD = a√3

ii)  Area(ΔABC)= 1/2 * BC * AD

= 1/2*2a*a√3

= a²√3​

Verified answer

To prove the given statements, let's analyze the given figure.

(i) Claim: AB = a√3

In a triangle with sides AB = BC = CA = 2a, it can be inferred that it is an equilateral triangle since all three sides are equal (2a = 2a = 2a). In an equilateral triangle, all angles are 60 degrees.

By drawing a perpendicular from point A to the side BC and labeling it as AD, we can create a right-angled triangle ADB, where angle BDA is 90 degrees.

In a right-angled triangle, we can use the Pythagorean theorem to relate the sides. In this case, the hypotenuse is AB and the other two sides are AD and BD.

Applying the Pythagorean theorem:

(AB)^2 = (AD)^2 + (BD)^2

Since angle BDA is 90 degrees, we can also determine that angle ADB is 30 degrees (as angle ADB + angle BDA = 180 degrees, and angle BDA = 90 degrees). Thus, triangle ADB is a 30-60-90 right triangle.

In a 30-60-90 right triangle, the sides are related by the following ratios:

AD = (1/2) * AB

BD = (sqrt(3)/2) * AB

Substituting these values into the Pythagorean theorem equation:

(AB)^2 = [(AD)^2 + (BD)^2]

(AB)^2 = [(1/2) AB]^2 + [(sqrt(3)/2) AB]^2

(AB)^2 = [(1/4) AB^2] + [(3/4) AB^2]

(AB)^2 = [(1/4) + (3/4)] * AB^2

(AB)^2 = AB^2

AB = a√3

Hence, we have proven that AB = a√3.

(ii) Claim: Area of ABC = a²√3

Since triangle ABC is an equilateral triangle, the area of an equilateral triangle can be calculated using the formula:

Area = (sqrt(3)/4) * (side)^2

Plugging in the values, we get:

Area = (sqrt(3)/4) * (2a)^2

Area = (sqrt(3)/4) * 4a^2

Area = a²√3

Therefore, we have proven that the area of triangle ABC is a²√3.

By following the above steps, we have established the given claims.

~Hope It helps.. ✨