Length of Perpendicular Drawn on Longest Side of Scalene Triangle

A scalene triangle is a triangle in which all three sides have different lengths. In this type of triangle, the longest side is generally referred to as the base. To find the length of the perpendicular drawn on the longest side of a scalene triangle, we can use the concept of the altitude.

Altitude of a Triangle:
The altitude of a triangle is a line segment drawn from a vertex of the triangle perpendicular to the opposite side or its extension. The length of the altitude is the perpendicular distance from the vertex to the opposite side.

Finding the Length of the Perpendicular:
To find the length of the perpendicular drawn on the longest side of a scalene triangle, we need to follow these steps:

1. Identify the longest side:
- In a scalene triangle, determine which side is the longest side. Let's call it side AB.
- Label the other two sides as AC and BC.

2. Find the altitude:
- Draw an altitude from the vertex opposite to the longest side (vertex C) to the longest side (AB).
- This altitude will be perpendicular to side AB.

3. Label the altitude:
- Let's call the point where the altitude intersects the longest side as D.
- Now we have a right triangle ADC, where AD is the altitude.

4. Use the Pythagorean theorem:
- Apply the Pythagorean theorem to the right triangle ADC to find the length of AD (the perpendicular).
- The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
- AD^2 + CD^2 = AC^2

5. Calculate the length of AD:
- Substitute the known values into the equation from step 4.
- Solve for AD by taking the square root of both sides of the equation.
- AD = √(AC^2 - CD^2)

6. Determine the length of the perpendicular:
- The length of the perpendicular drawn on the longest side of the scalene triangle is equal to the calculated length of AD.

Summary:
To find the length of the perpendicular drawn on the longest side of a scalene triangle, follow these steps:
1. Identify the longest side.
2. Find the altitude by drawing a perpendicular line from the opposite vertex to the longest side.
3. Label the point where the altitude intersects the longest side.
4. Use the Pythagorean theorem to calculate the length of the perpendicular.
5. Determine the length of the perpendicular by taking the square root of the calculated value.

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