PQR is an isosceles triangle with sides `PQ = PR = 45 cm `and `QR = 72 cm`. PN is a median to base QR. What will be the length (in cm.) of PN?

Options

A. 36
B. 24
C. 27
D. 32

Answer

To find the length of the median PN in the isosceles triangle PQR, we can follow these steps:

### Step 1: Understand the triangle and the median
Given triangle PQR is isosceles with sides PQ = PR = 45 cm and base QR = 72 cm. The median PN divides the base QR into two equal parts, meaning QN = NR = 36 cm (since QR = 72 cm).

### Step 2: Use the Pythagorean theorem
In triangle PQN, we can apply the Pythagorean theorem. Here, PN is the height from point P to base QR, and QN is half of QR.

### Step 3: Set up the equation
Let PN = x. According to the Pythagorean theorem:
\[ PQ^2 = PN^2 + QN^2 \]
Substituting the known values:
\[ 45^2 = x^2 + 36^2 \]

### Step 4: Calculate the squares
Calculate \( 45^2 \) and \( 36^2 \):
- \( 45^2 = 2025 \)
- \( 36^2 = 1296 \)

### Step 5: Substitute and solve for x
Now substitute these values into the equation:
\[ 2025 = x^2 + 1296 \]
Rearranging gives:
\[ x^2 = 2025 - 1296 \]
\[ x^2 = 729 \]

### Step 6: Find the value of x
Taking the square root of both sides:
\[ x = \sqrt{729} = 27 \text{ cm} \]

### Conclusion
Thus, the length of the median PN is **27 cm**.

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