Question

right triangle similarity
what is the value of x?
3 units
2 units
8 units
5 units
(image of a right triangle with right angle at b, altitude bd of length 10 to hypotenuse ac, segment ad labeled x, segment dc labeled 4x)

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. So, \( BD^2 = AD \times DC \).
Given \( BD = 10 \), \( AD = x \), \( DC = 4x \), we substitute: \( 10^2 = x \times 4x \).

Step2: Solve the Equation

Simplify the equation: \( 100 = 4x^2 \). Divide both sides by 4: \( x^2 = \frac{100}{4} = 25 \). Take the square root: \( x = \sqrt{25} = 5 \)? Wait, no, wait—wait, maybe I made a mistake. Wait, no, the options have 5? Wait, the options are 3,2,8,5. Wait, let's recheck. Wait, the Geometric Mean Theorem: In right triangle \( ABC \) with right angle at \( B \), and altitude \( BD \) to hypotenuse \( AC \), then \( BD^2 = AD \times DC \), \( AB^2 = AD \times AC \), \( BC^2 = DC \times AC \). Wait, maybe I misapplied. Wait, \( BD = 10 \), \( AD = x \), \( DC = 4x \). So \( BD^2 = AD \times DC \) → \( 10^2 = x \times 4x \) → \( 100 = 4x^2 \) → \( x^2 = 25 \) → \( x = 5 \). But wait, the options have 5 units. Wait, but let me check again. Wait, maybe the diagram is different? Wait, the triangle is right-angled at \( B \), \( BD \) is altitude to \( AC \), so yes, the geometric mean theorem (altitude-on-hypotenuse theorem) states that the altitude is the geometric mean of the two segments. So \( BD = \sqrt{AD \times DC} \), so \( 10 = \sqrt{x \times 4x} = \sqrt{4x^2} = 2x \). Oh! Wait, that's the mistake. I squared \( BD \), but actually, \( BD = \sqrt{AD \times DC} \), so \( 10 = \sqrt{x \times 4x} = \sqrt{4x^2} = 2x \). Then solving \( 2x = 10 \) → \( x = 5 \)? Wait, no, \( 2x = 10 \) → \( x = 5 \)? Wait, but let's do that again. \( AD = x \), \( DC = 4x \), so \( AD \times DC = x \times 4x = 4x^2 \). Then \( BD = \sqrt{4x^2} = 2x \). So \( 2x = 10 \) → \( x = 5 \). Yes, that's correct. So \( x = 5 \). So the answer is 5 units.

Answer:

5 units (the option with "5 units")