Triangle ABC

Given:
- Triangle ABC with sides AB = 2 and AC = 3.
- Median AD is perpendicular to side BC.

Approach

To find the area of triangle ABC, we can use the formula for the area of a triangle: Area = (base * height) / 2. In this case, we need to find the base and height of the triangle.

Finding the Base

Since AD is a median, it divides BC into two equal parts. Let's denote the point of intersection of AD and BC as E. Therefore, BE = EC.

Finding the Height

The height of the triangle can be found by drawing a perpendicular line from A to BC. Let's denote the point of intersection as F. Therefore, AF is the height of the triangle.

Finding the Length of BE and EC

Since AD is a median, it divides BC into two equal parts. Therefore, BE = EC = BC/2.

Finding the Length of AF

To find the length of AF, we can use the Pythagorean theorem. In triangle ACF, AC = 3 and FC = BC/2. Using the Pythagorean theorem, we get:

AF^2 = AC^2 - FC^2
AF^2 = 3^2 - (BC/2)^2
AF^2 = 9 - (BC^2/4)
AF^2 = (36 - BC^2)/4

Finding the Length of BC

To find the length of BC, we can use the Pythagorean theorem. In triangle ABC, AB = 2 and AC = 3. Using the Pythagorean theorem, we get:

BC^2 = AB^2 + AC^2
BC^2 = 2^2 + 3^2
BC^2 = 4 + 9
BC^2 = 13

Finding the Height AF

Substituting BC^2 = 13 in the equation we derived earlier for AF^2, we get:

AF^2 = (36 - 13)/4
AF^2 = 23/4

Taking the square root of both sides, we get:

AF = sqrt(23)/2

Calculating the Area of Triangle ABC

Now that we have the base BE/EC = BC/2 = sqrt(13)/2 and the height AF = sqrt(23)/2, we can use the formula for the area of a triangle to calculate the area of triangle ABC:

Area = (base * height) / 2
Area = (sqrt(13)/2 * sqrt(23)/2) / 2
Area = (sqrt(299))/4

Therefore, the area of triangle ABC is (sqrt(299))/4.