A circle is inscribed in a triangle ABC, having sides 8cm, 10cm and 12cm. Find AD, BE and CF (these 3 are altitudes of triangle ABC).

To find the altitudes AD, BE, and CF of triangle ABC with sides 8 cm, 10 cm, and 12 cm, we first need to calculate the area of the triangle using Heron's formula.

Step 1: Calculate the Semi-Perimeter

The semi-perimeter (s) of the triangle is given by:

• s = (a + b + c) / 2
• s = (8 + 10 + 12) / 2 = 15 cm

Step 2: Calculate the Area

Using Heron's formula, the area (A) can be calculated as follows:

• A = √(s × (s - a) × (s - b) × (s - c))
• A = √(15 × (15 - 8) × (15 - 10) × (15 - 12))
• A = √(15 × 7 × 5 × 3) = √(1575) = 15√7 cm²

Step 3: Calculate the Altitudes

The altitude can be found using the formula:

• Altitude = (2 × Area) / Base

Finding AD (altitude from A to BC)

Using side BC (10 cm) as the base:

• AD = (2 × 15√7) / 10 = 3√7 cm

Finding BE (altitude from B to AC)

Using side AC (12 cm) as the base:

• BE = (2 × 15√7) / 12 = 5√7 / 2 cm

Finding CF (altitude from C to AB)

Using side AB (8 cm) as the base:

• CF = (2 × 15√7) / 8 = (15√7) / 4 cm

Final Results

The altitudes of triangle ABC are:

• AD = 3√7 cm
• BE = 5√7 / 2 cm
• CF = (15√7) / 4 cm