ABC is a triangle right angled at B. The hypotenuse (AC) is four times the perpendicular (BD) drawn to it from the opposite vertex and `ADltDC`.
What is tan(A-C) equal to?

Options

A. A. 0
B. B. 1
C. C. 2
D. D. none

Answer

To solve the problem step by step, we will analyze the triangle ABC, which is right-angled at B, and use the given information to find the value of tan(A - C).

### Step 1: Understand the Triangle Configuration
We have triangle ABC, where:
- Angle B is 90 degrees.
- AC is the hypotenuse.
- BD is the perpendicular from B to AC.

### Step 2: Define Variables
Let BD = X (the length of the perpendicular). According to the problem, the hypotenuse AC is four times the length of BD:
\[ AC = 4X \]

### Step 3: Identify Angles
Let angle ACB = θ. Therefore:
- Angle A = A
- Angle C = 90° - θ (since the sum of angles in a triangle is 180°).

### Step 4: Use Trigonometric Ratios
In triangle ABC:
- The cosine of angle θ (cos θ) is given by the ratio of the adjacent side (BC) to the hypotenuse (AC):
\[ \cos θ = \frac{BC}{AC} = \frac{BC}{4X} \]

In triangle BDC:
- The sine of angle θ (sin θ) is given by the ratio of the opposite side (BD) to the hypotenuse (BC):
\[ \sin θ = \frac{BD}{BC} = \frac{X}{BC} \]

### Step 5: Multiply the Two Equations
From the two equations we have:
1. \( \cos θ = \frac{BC}{4X} \)
2. \( \sin θ = \frac{X}{BC} \)

Multiplying these equations:
\[ \sin θ \cdot \cos θ = \frac{X}{BC} \cdot \frac{BC}{4X} \]
This simplifies to:
\[ \sin θ \cdot \cos θ = \frac{1}{4} \]

### Step 6: Use the Double Angle Identity
Using the identity \( 2 \sin θ \cos θ = \sin 2θ \):
\[ 2 \sin θ \cos θ = \frac{1}{2} \]
Thus:
\[ \sin 2θ = \frac{1}{2} \]

### Step 7: Solve for θ
The value of \( 2θ \) for which \( \sin 2θ = \frac{1}{2} \) is:
\[ 2θ = 30° \quad \text{or} \quad 2θ = 150° \]
Thus:
\[ θ = 15° \quad \text{or} \quad θ = 75° \]
Since angle C is less than angle A, we take \( θ = 15° \).

### Step 8: Find Angle A
Using the angle sum property in triangle ABC:
\[ A + C + 90° = 180° \]
So:
\[ A + (90° - θ) = 90° \]
Thus:
\[ A = 90° - 15° = 75° \]

### Step 9: Calculate tan(A - C)
Now we have:
- Angle A = 75°
- Angle C = 15°

We need to find:
\[ \tan(A - C) = \tan(75° - 15°) = \tan(60°) \]

### Step 10: Evaluate tan(60°)
We know that:
\[ \tan(60°) = \sqrt{3} \]

### Final Answer
Thus, the value of \( \tan(A - C) \) is:
\[ \tan(A - C) = \sqrt{3} \]

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