In the following figures AB = 4 m and ED = 3m.
If ` sin alpha = (3)/(5) and cos beta = (12)/(13) ` find the length of BD

Answer

To find the length of BD, we will break the problem down into steps involving triangles ABC and CED.

### Step 1: Identify the given values
- AB = 4 m
- ED = 3 m
- sin(α) = 3/5
- cos(β) = 12/13

### Step 2: Calculate the length of BC using triangle ABC
From the definition of sine:
\[ \sin(α) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{3}{5} \]
Here, the opposite side (perpendicular) is 3, and the hypotenuse is 5.

Using the Pythagorean theorem:
\[ \text{Base}^2 + \text{Perpendicular}^2 = \text{Hypotenuse}^2 \]
Let the base be B. Then:
\[ B^2 + 3^2 = 5^2 \]
\[ B^2 + 9 = 25 \]
\[ B^2 = 16 \]
\[ B = \sqrt{16} = 4 \]

### Step 3: Calculate BC using tan(α)
Using the definition of tangent:
\[ \tan(α) = \frac{\text{Opposite}}{\text{Base}} = \frac{3}{4} \]
In triangle ABC:
\[ \tan(α) = \frac{AB}{BC} = \frac{4}{BC} \]
Setting the two equal:
\[ \frac{4}{BC} = \frac{3}{4} \]
Cross-multiplying gives:
\[ 4 \cdot 4 = 3 \cdot BC \]
\[ 16 = 3 \cdot BC \]
\[ BC = \frac{16}{3} \text{ m} \]

### Step 4: Calculate the length of CD using triangle CED
From the definition of cosine:
\[ \cos(β) = \frac{\text{Base}}{\text{Hypotenuse}} = \frac{12}{13} \]
Here, the base is 12, and the hypotenuse is 13.

Using the Pythagorean theorem:
\[ \text{Perpendicular}^2 + \text{Base}^2 = \text{Hypotenuse}^2 \]
Let the perpendicular be P. Then:
\[ P^2 + 12^2 = 13^2 \]
\[ P^2 + 144 = 169 \]
\[ P^2 = 25 \]
\[ P = \sqrt{25} = 5 \]

### Step 5: Calculate CD using tan(β)
Using the definition of tangent:
\[ \tan(β) = \frac{\text{Opposite}}{\text{Base}} = \frac{5}{12} \]
In triangle CED:
\[ \tan(β) = \frac{ED}{CD} = \frac{3}{CD} \]
Setting the two equal:
\[ \frac{3}{CD} = \frac{5}{12} \]
Cross-multiplying gives:
\[ 3 \cdot 12 = 5 \cdot CD \]
\[ 36 = 5 \cdot CD \]
\[ CD = \frac{36}{5} \text{ m} \]

### Step 6: Find the length of BD
Now, we can find BD by adding BC and CD:
\[ BD = BC + CD = \frac{16}{3} + \frac{36}{5} \]
To add these fractions, we need a common denominator. The LCM of 3 and 5 is 15.
\[ BD = \frac{16 \cdot 5}{15} + \frac{36 \cdot 3}{15} = \frac{80}{15} + \frac{108}{15} = \frac{188}{15} \text{ m} \]

### Step 7: Convert to mixed fraction
\[ \frac{188}{15} = 12 \frac{8}{15} \text{ m} \]

Thus, the length of BD is:
\[ \text{BD} = 12 \frac{8}{15} \text{ m} \]