Question

what is the value of x? triangle diagram with points a, b, c, d; ad = x, dc = 4x, bd = 10, right angle at d and at b. multiple choice options: 2 units, 5 units, 8 units, 3 units

Explanation:

Step1: Apply Geometric Mean Theorem

In a right triangle, the altitude to the hypotenuse is the geometric mean of the segments into which it divides the hypotenuse. Also, each leg is the geometric mean of the hypotenuse and the adjacent segment. Here, \( BD = 10 \), \( AD=x \), \( DC = 4x \). By the geometric mean theorem for right triangles (altitude-on-hypotenuse theorem), we have \( BD^{2}=AD\times DC \).
So, \( 10^{2}=x\times4x \).

Step2: Solve the Equation

Simplify the equation: \( 100 = 4x^{2} \).
Divide both sides by 4: \( \frac{100}{4}=x^{2} \), which gives \( 25=x^{2} \).
Take the square root of both sides: \( x = \sqrt{25}=5 \) (we consider the positive root since length can't be negative).

Answer:

5