The sides of a scalene triangle are 11cm ,12cm and 13 cm . The length of thealtitude corresponding to the side having length is 12 CM

To find the length of the altitude corresponding to the side of a scalene triangle with a length of 12 cm, we can use a combination of Heron's formula and the formula for the area of a triangle. Let's break this down step by step.

Step 1: Calculate the Semi-Perimeter

First, we need to determine the semi-perimeter (s) of the triangle. The semi-perimeter is half the sum of all the sides. For our triangle with sides 11 cm, 12 cm, and 13 cm, we calculate:

• s = (11 cm + 12 cm + 13 cm) / 2
• s = 36 cm / 2 = 18 cm

Step 2: Apply Heron's Formula

Next, we can use Heron's formula to find the area (A) of the triangle. Heron's formula states that:

A = √(s × (s - a) × (s - b) × (s - c))

where a, b, and c are the lengths of the sides of the triangle. Plugging in our values:

• A = √(18 cm × (18 cm - 11 cm) × (18 cm - 12 cm) × (18 cm - 13 cm))
• A = √(18 cm × 7 cm × 6 cm × 5 cm)

Calculating this gives:

• A = √(18 × 7 × 6 × 5)
• A = √(3780) ≈ 61.5 cm²

Step 3: Calculate the Altitude

Now that we have the area of the triangle, we can find the altitude (h) corresponding to the side of length 12 cm using the formula for the area of a triangle:

A = (1/2) × base × height

Here, the base is 12 cm, and we want to find the height (altitude) corresponding to this base:

• 61.5 cm² = (1/2) × 12 cm × h

To isolate h, we rearrange the equation:

• h = (2 × 61.5 cm²) / 12 cm
• h = 123 cm² / 12 cm
• h = 10.25 cm

Final Result

The length of the altitude corresponding to the side of length 12 cm in the scalene triangle is approximately 10.25 cm. This method effectively combines geometry and algebra to solve for the desired measurement, illustrating the interconnectedness of these mathematical concepts.