From the vertex A of a right angled triangle ABC, (A being the right angle) AD is drawn perpendicular to BC. How do I prove that the three triangles ABD, ADC, and ABC are similar?

Thank you!

To prove that the triangles ABD, ADC, and ABC are similar, we need to show that their corresponding angles are equal and their corresponding sides are proportionate.

First, let's establish the corresponding angles between the triangles:

1. Angle ABD and angle ABC: Since AD is perpendicular to BC, angle ABD is a right angle (90 degrees). Angle ABC is also a right angle since it is given that triangle ABC is a right-angled triangle. Hence, angle ABD and angle ABC are both right angles and therefore equal.

2. Angle ADC and angle ABC: Since AD is perpendicular to BC, angle ADC is also a right angle (90 degrees), making it equal to angle ABC.

Now, let's consider the corresponding sides of the triangles:

1. Side AB and side AC: These two sides are equal in length because they form the legs of the right-angled triangle ABC. This can be proven using the Pythagorean theorem or other methods specific to the given triangle.

2. Side AD: This is the common hypotenuse for triangles ABD and ADC. By definition, the hypotenuse is the longest side of a right-angled triangle. Therefore, side AD will be the longest side for both ABD and ADC.

Since we have established that the corresponding angles are equal and the corresponding sides are proportionate, we can conclude that triangles ABD, ADC, and ABC are similar.