AD is perpendicular to BC

Question

AD is perpendicular to BC

AB=AC=a ,AD=x

 Find area of triangle ABC

 Options:-

 (a) 2x(a^2--x^2)^1/2

 (b) x(a^2--x^2)^1/2

 (c) 2a(a^2--x^2)^1/2

 (d) a(a^2--x^2)^1/2

 plz answer with explaination 

Answer 1

Try drawing yourself a picture of triangle ABC with line AD perpendicular to BC.  For a triangle:

 

      Area = (1/2)*base*height

 

where the height = AD = x and the base = BC

 

      Area = (1/2)*BC*x

 

Now, since AB=AC = a, the triangle is equilateral, so CD = DB = (1/2)BC and AD is perpendicular to BC.  Hence  triangle ACD is a right triangle.  By the Pythagorean Theorem CD²+x² = a², so CD² = a² - x² and CD = (a²-x²)^1/2.  Finally

 

      Area = (1/2)BC*height = CD*x = (a²-x²)^1/2*x

 

The answer is (b)