In a triangle ABC, the lengths of the sides AB and AC equal to 17.5 cm an 9 cm respectively . Let D be a point on the line segment BC such that AD is perpendicular to BC. If AD=3 cm, the what is the radius (in cm ) of the circule circumscribing the triangle ABC ?

Answer

To find the radius of the circumcircle of triangle ABC, we will follow these steps:

### Step 1: Identify the given values
- Length of side AB (denote as \( b \)) = 17.5 cm
- Length of side AC (denote as \( c \)) = 9 cm
- Length of altitude AD = 3 cm

### Step 2: Determine the length of side BC (denote as \( a \))
Since we don't have the length of side BC, we will use the right triangle formed by altitude AD. We can apply the Pythagorean theorem. Let \( BD = x \) and \( DC = y \), then \( BC = a = x + y \).

Using the Pythagorean theorem on triangles ABD and ACD:
1. For triangle ABD:
\[ AB^2 = AD^2 + BD^2 \implies 17.5^2 = 3^2 + x^2 \]
\[ 306.25 = 9 + x^2 \implies x^2 = 306.25 - 9 = 297.25 \implies x = \sqrt{297.25} \]

2. For triangle ACD:
\[ AC^2 = AD^2 + DC^2 \implies 9^2 = 3^2 + y^2 \]
\[ 81 = 9 + y^2 \implies y^2 = 81 - 9 = 72 \implies y = \sqrt{72} \]

### Step 3: Calculate the lengths \( x \) and \( y \)
- \( x = \sqrt{297.25} \approx 17.24 \) cm
- \( y = \sqrt{72} \approx 8.49 \) cm

### Step 4: Calculate the length of side BC
\[ BC = a = x + y = \sqrt{297.25} + \sqrt{72} \approx 17.24 + 8.49 \approx 25.73 \text{ cm} \]

### Step 5: Calculate the area of triangle ABC
Using the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, we can take \( BC \) as the base and \( AD \) as the height:
\[ \text{Area} = \frac{1}{2} \times a \times AD = \frac{1}{2} \times 25.73 \times 3 \approx 38.595 \text{ cm}^2 \]

### Step 6: Calculate the radius \( R \) of the circumcircle
Using the formula:
\[ R = \frac{abc}{4 \times \text{Area}} \]
Where \( a = 25.73 \), \( b = 17.5 \), and \( c = 9 \):
\[ R = \frac{25.73 \times 17.5 \times 9}{4 \times 38.595} \]
Calculating the numerator:
\[ 25.73 \times 17.5 \times 9 \approx 4056.825 \]
Calculating the denominator:
\[ 4 \times 38.595 \approx 154.38 \]
Now, calculate \( R \):
\[ R \approx \frac{4056.825}{154.38} \approx 26.29 \text{ cm} \]

### Final Answer:
The radius of the circumcircle of triangle ABC is approximately **26.29 cm**.

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