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 `angleABD?`

Options

A. A. `15^(@)`
B. B. `30^(@)`
C. C. `45^(@)`
D. D. none

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step 1: Understand the Triangle
We have a triangle ABC that is right-angled at B. The hypotenuse AC is four times the length of the perpendicular BD drawn from B to AC.

### Step 2: Define Variables
Let BD = X. Then, according to the problem, the hypotenuse AC = 4 * BD = 4X.

### Step 3: Identify Angles
Let angle ACB = θ. We need to find angle ABD.

### Step 4: Use Cosine in Triangle ABC
In triangle ABC, we can use the cosine function:
\[ \cos(θ) = \frac{BC}{AC} = \frac{BC}{4X} \]
This gives us our first equation:
\[ BC = 4X \cos(θ) \quad \text{(1)} \]

### Step 5: Use Sine in Triangle BDC
In triangle BDC, we can use the sine function:
\[ \sin(θ) = \frac{BD}{BC} = \frac{X}{BC} \]
This gives us our second equation:
\[ BC = \frac{X}{\sin(θ)} \quad \text{(2)} \]

### Step 6: Equate the Two Expressions for BC
From equations (1) and (2), we have:
\[ 4X \cos(θ) = \frac{X}{\sin(θ)} \]
Canceling X (assuming X ≠ 0), we get:
\[ 4 \cos(θ) = \frac{1}{\sin(θ)} \]
Rearranging gives:
\[ \sin(θ) = \frac{1}{4 \cos(θ)} \]

### Step 7: Multiply Both Sides by 4
Multiplying both sides by 4 gives:
\[ 4 \sin(θ) \cos(θ) = 1 \]
Using the double angle identity \( \sin(2θ) = 2 \sin(θ) \cos(θ) \), we can rewrite this as:
\[ 2 \sin(2θ) = 1 \]
Thus,
\[ \sin(2θ) = \frac{1}{2} \]

### Step 8: Solve for 2θ
The sine function equals \( \frac{1}{2} \) at \( 2θ = 30^\circ \) or \( 2θ = 150^\circ \). However, since θ must be an angle in a triangle, we take:
\[ 2θ = 30^\circ \Rightarrow θ = 15^\circ \]

### Step 9: Find Angle A
In triangle ABC, the sum of angles is 180°:
\[ \angle A + \angle B + \angle C = 180° \]
Given that angle B = 90° and angle C = 15°:
\[ \angle A + 90° + 15° = 180° \]
This simplifies to:
\[ \angle A = 180° - 105° = 75° \]

### Step 10: Find Angle ABD
Now we will use triangle ABD to find angle ABD:
\[ \angle ABD + \angle A + \angle B = 180° \]
Substituting the known values:
\[ \angle ABD + 75° + 90° = 180° \]
This simplifies to:
\[ \angle ABD + 165° = 180° \]
Thus,
\[ \angle ABD = 180° - 165° = 15° \]

### Final Answer
The value of angle ABD is \( 15^\circ \).

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