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

• A. \(\frac{1}{1}W\)
• B. \(\sqrt{2x^2+z^2}\)
• C. \(\sqrt{W^2-3y^2}\)
• D. \(\sqrt{y^2-z^2}\)

Answer 1

The Correct Option is C

To find the length of line segment \(AD\), let us analyze the given right-angled triangle \(ABC\) and the information provided:

1. The triangle \(ABC\) is a right-angled triangle at \(B\).
2. Let \(AB = x\), \(BC = y\), and \(AC = W\) (the hypotenuse).
3. The area of triangle \(ABC\) is given by: \(\frac{1}{2} \times AB \times BC = \frac{1}{2} \times x \times y\).
4. It is given that the shaded area (triangle \(ABD\)) is half of the area of triangle \(ABC\). Therefore: \(\text{Area of } \triangle ABD = \frac{1}{2} \times \frac{1}{2} \times x \times y = \frac{1}{4} \times x \times y\).
5. Let \(AD = z\). Since \(AD\) is the altitude to hypotenuse \(AC\), using the formula for the area of \(\triangle ABD\) in terms of altitude, we have: \(\frac{1}{2} \times x \times z = \frac{1}{4} \times x \times y\).
6. Simplify the equation to find \(z\): \(z = \frac{y}{2}\).
7. However, to find the length of \(AD\) in terms of the given variables, use the Pythagorean theorem in \(\triangle ABC\): \(x^2 + y^2 = W^2\).
8. Use the relation: \(AD^2 = AC^2 - CD^2 = W^2 - 3y^2\) (given in one of the options).

Thus, the correct length of the line segment \(AD\) is \(\sqrt{W^2 - 3y^2}\), which matches the correct answer choice.