ABC is an isosceles triangle such that `AB = AC = 30 cm `and `BC = 48 cm`. AD is a median to base BC. What is the length (in cm) of AD?

Options

A. 18
B. 20
C. 24
D. 32

Answer

To find the length of the median \( AD \) in the isosceles triangle \( ABC \) where \( AB = AC = 30 \, \text{cm} \) and \( BC = 48 \, \text{cm} \), we can follow these steps:

### Step 1: Identify the lengths of the segments
Since \( D \) is the midpoint of \( BC \), we can find the lengths of \( BD \) and \( DC \):
\[ BD = DC = \frac{BC}{2} = \frac{48 \, \text{cm}}{2} = 24 \, \text{cm} \]

### Step 2: Apply the Pythagorean theorem
In triangle \( ABD \), we can apply the Pythagorean theorem since \( AD \) is the median and is perpendicular to \( BC \):
\[ AB^2 = AD^2 + BD^2 \]
Substituting the known values:
\[ 30^2 = AD^2 + 24^2 \]

### Step 3: Calculate the squares
Calculating the squares:
\[ 900 = AD^2 + 576 \]

### Step 4: Solve for \( AD^2 \)
Rearranging the equation to solve for \( AD^2 \):
\[ AD^2 = 900 - 576 \]
\[ AD^2 = 324 \]

### Step 5: Find \( AD \)
Taking the square root of both sides gives:
\[ AD = \sqrt{324} = 18 \, \text{cm} \]

### Conclusion
The length of the median \( AD \) is \( 18 \, \text{cm} \).

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