Let ABC be a right triangle with length of side `AB=3` and hyotenus `AC=5.` If D is a point on BC such that `(BD)/(DC)=(AB)/(AC),` then AD is equal to

Options

A. ` (4sqrt3)/(3)`
B. `(3sqrt5)/(2)`
C. `(4 sqrt5)/(3)`
D. `(5 sqrt3)/(4)`

Answer

To solve the problem step-by-step, we will follow the logic presented in the video transcript.

### Step 1: Identify the triangle and given values
We have a right triangle \( ABC \) where:
- \( AB = 3 \) (one leg)
- \( AC = 5 \) (hypotenuse)

### Step 2: Find the length of side \( BC \)
Using the Pythagorean theorem:
\[ AB^2 + BC^2 = AC^2 \]
Substituting the known values:
\[ 3^2 + BC^2 = 5^2 \]
\[ 9 + BC^2 = 25 \]
\[ BC^2 = 25 - 9 = 16 \]
\[ BC = \sqrt{16} = 4 \]

### Step 3: Set up the ratio \( \frac{BD}{DC} \)
We are given that:
\[ \frac{BD}{DC} = \frac{AB}{AC} = \frac{3}{5} \]
Let \( BD = 3x \) and \( DC = 5x \). Then, we can express \( BC \) as:
\[ BD + DC = BC \implies 3x + 5x = 4 \implies 8x = 4 \implies x = \frac{1}{2} \]
Thus:
\[ BD = 3x = 3 \cdot \frac{1}{2} = \frac{3}{2} \]
\[ DC = 5x = 5 \cdot \frac{1}{2} = \frac{5}{2} \]

### Step 4: Find \( AD \) using the Pythagorean theorem in triangle \( ABD \)
Now we will use the Pythagorean theorem again in triangle \( ABD \):
\[ AD^2 = AB^2 + BD^2 \]
Substituting the known values:
\[ AD^2 = 3^2 + \left(\frac{3}{2}\right)^2 \]
\[ AD^2 = 9 + \frac{9}{4} \]
To combine these, convert \( 9 \) into a fraction:
\[ 9 = \frac{36}{4} \]
Thus:
\[ AD^2 = \frac{36}{4} + \frac{9}{4} = \frac{45}{4} \]
Taking the square root:
\[ AD = \sqrt{\frac{45}{4}} = \frac{\sqrt{45}}{2} = \frac{3\sqrt{5}}{2} \]

### Final Answer
Thus, the length of \( AD \) is:
\[ AD = \frac{3\sqrt{5}}{2} \]