Question

In the following diagram, if the shaded area is one half the area of triangle ABC and angle ABC is right angle then the length of line segment AD is

        C
       /|
      / | z
     /  |
 W  /   D
   /   /|
  /   / | y
 /   /  |
A-------B
    x

(Note: $AC = W$ , $AB = x$ , $BD = y$ , $DC = z$ )

Accepted Answer

C. Area of $	riangle ABC = \frac{1}{2} x (y + z)$. Area of $	riangle ABD = \frac{1}{2} xy$. Given shaded area ($	riangle ACD$) is half the area of $	riangle ABC$, so Area($	riangle ABC$) $= 2 	imes$ Area($	riangle ABD$): $$\frac{1}{2} x (y + z) = 2 \left(\frac{1}{2} xyight) \Rightarrow y + z = 2y \Rightarrow z = y$$ In right $	riangle ABC$: $$W^2 = x^2 + (y + z)^2 = x^2 + (2y)^2 = x^2 + 4y^2$$ In right $	riangle ABD$: $$AD^2 = x^2 + y^2 = (W^2 - 4y^2) + y^2 = W^2 - 3y^2$$ $$AD = \sqrt{W^2 - 3y^2}$$

Geometry of Right Angled Triangle Area

Explanation

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