Calculating the Height from Point B to Side AC in Triangle ABC

In triangle ABC, where AB 7 cm, BC 5 cm, and AC 6 cm, the height AD from A to BC is 10 cm. To find the height BE from point B to side AC, we can utilize the area of the triangle in two different ways.

Step 1: Calculating the Area Using Height AD

The area A of triangle ABC can be calculated using the base BC and height AD as follows:

A 1/2 × base × height 1/2 × BC × AD

Substituting the given values:

A 1/2 × 5 cm × 10 cm 1/2 × 50 cm2 25 cm2

Step 2: Calculating the Area Using Height BE

Alternatively, we can calculate the area using the base AC and height BE as:

A 1/2 × base × height 1/2 × AC × BE

Substituting the known values:

25 cm2 1/2 × 6 cm × BE

Step 3: Solving for Height BE

Now, we can solve for BE:

25 3 × BE

BE 25 / 3 ≈ 8.33 cm

Thus, the height BE from point B to side AC is approximately 8.33 cm.

Conclusion

The height BE from point B to side AC has been calculated to be approximately 8.33 cm using the relationship between the area of the triangle and the heights from different vertices.

Note: If AD 10 cm, it contradicts the triangle inequality theorem, as the sum of any two sides of the triangle must be greater than the third side. However, for the calculation of the height BE, the area method consistently holds true.