1. Find the length of segment CD 2. Find the length of segment AD. 3. What is the similarity factor between △ABC & △CBD? 4. What is the ratio of the area of △ to the area of △CBD?

If an altitude is drawn from the right angle of any right triangle, then the two triangles formed are similar to the original triangle and all three triangles are similar to each other.

If two objects are similar, their corresponding angles are congruent (the same) and their sides are proportional in length.

Therefore, if m∠B = 60° then m∠C of ΔADC = 60°

If m∠A = 30° then m∠C of ΔCBD = 30°

This means that all triangles are 30-60-90 triangles. 30-60-90 triangles are special right triangles as the angles of these triangles are in a unique ratio of 1:2:3. This means that the lengths of the sides is always in the ratio of 1 : √3 : 2 ⇒ y : y√3 : 2y

y: side opposite 30° angle
y√3: side opposite 60° angle
2y: hypotenuse

1. Using the 30-60-90 triangle formula for sides:

For ΔCBD: CD = y√3, CB = 2y, DB = y

We are told CB = 12 ⇒ 2y = 12 ⇒ y = 6

Therefore, CD = y√3 = 6√3

2. Using the 30-60-90 triangle formula for sides:

For ΔADC: CD = y, AC = 2y, AD = y√3

We know that CD = 6√3 ⇒ y = 6√3

Therefore, AD = y√3 = 6√3√3 = 18

3. Using the 30-60-90 triangle formula for sides:

For ΔABC: CB = y, AB = 2y, AC = y√3

We know that CB = 12 ⇒ y = 12

Therefore, AB = 24 and AC = 12√3

Therefore, ratio of ΔABC : ΔCBD using the lengths of their hypotenuse:

24 : 12 ⇒ 2 : 1

Therefore, triangle ΔABC is an enlargement of triangle ΔCBD by a scale factor of 2

4. Area of a triangle = 1/2 x base x height

⇒ Area ΔABC = 1/2 x BC x AC = 1/2 x 12 x 12√3 = 72√3

⇒ Area ΔCBD = 1/2 x DB x CD = 1/2 x 6 x 6√3 = 18√3

Therefore, Area ΔABC : Area ΔCBD ⇒ 72√3 : 18√3 ⇒ 4 : 1