a) Triangle ADC ~ Triangle ABC (AA Similarity)
Triangle ADC ~ Triangle BDC (AA Similarity)
b) The lengths of the segments are: AC = 5
CB = 5
DB = 4 .
a) Similar Triangles
Triangles are similar if they have the same shape but different sizes.
We can see that triangles ADC and ABC are similar because they share two angles:
Angle A: Both triangles have a right angle at C.
Angle D: This is the angle between AC and CD in triangle ADC, and also between BC and AB in triangle ABC.
Therefore, we can write the following proportion:
ADC ~ ABC
We can find another similar triangle by looking at triangle BDC and triangle ACD. These two triangles share two angles as well:
Angle D: As mentioned before, this is the angle between AC and CD in triangle ADC, and also between BC and AB in triangle BDC.
Angle B: This is the right angle in both triangles.
Therefore, we can write another proportion:
BDC ~ ACD
b) Finding the lengths of AC, CB, and DB:
We can use the Pythagorean Theorem to find the length of AC:
AC^2 = BC^2 + AB^2
AC^2 = 3^2 + 4^2
AC^2 = 9 + 16
AC^2 = 25
AC = 5
Once we have the length of AC, we can use the ratios from the similar triangles to find the lengths of CB and DB:
ADC ~ ABC
AD/AC = AB/BC
4/5 = 4/BC
BC = 5
ADC ~ BDC
AD/AC = BD/CB
4/5 = BD/5
BD = 4
Therefore, the lengths of the segments are:
AC = 5
CB = 5
DB = 4