Short Answer 
Similar triangles maintain the same shape with equal corresponding angles and constant side length ratios, determinable by criteria like AA, SSS, and SAS. In the problem, triangles DAC and DBC are similar, allowing the calculation of unknown side lengths using established ratios, resulting in (DC=100) and (AD=64).
Understanding Similar Triangles
Similar triangles are triangles that have the same shape but can differ in size. This means that their corresponding angles are equal, and the ratios of their corresponding side lengths are constant. To determine if triangles are similar, you can use criteria such as Angle-Angle (AA), Side-Side-Side (SSS), or Side-Angle-Side (SAS) similarity.
Applying Similar Triangles Property
In the problem given, we establish that triangle DAC is similar to triangle DBC based on the condition √C¬†BD √¢¬a¬• AC. The property of similar triangles states that the ratios of the lengths of corresponding sides are equal. Specifically, we can write the equations to calculate unknown side lengths:
- (frac{AB}{BC}=frac{DB}{DC})
- (frac{DB}{DC}=frac{AD}{BD})
Calculating Lengths Using Ratios
Once we set up the equal ratios, we can solve for the unknown lengths (AD) and (DC). For instance, from the equations, we can derive (DC) as follows: (DC=frac{80times5}{4}) which results in (DC=100). Similarly, for (AD), we can use the ratio to find (AD= frac{80times80}{100}), leading us to (AD=64).