Notice that angle B corresponds to one of the interior angles of triangle ADB and triangle CAB, since both triangles are right triangles, then by AA criterion
[tex]\Delta ADB\approx\Delta CAB.[/tex]
Now, we are given that the length of segment AC is 8 units and the length of segment AB is 10 units, since the triangles are right triangles, from the Pythagorean theorem we get that:
[tex]BC^2+AC^2=AB^2.[/tex]
Solving the above equation for BC, we get:
[tex]BC=\sqrt{AB^2-AC^2}.[/tex]
Therefore:
[tex]BC=\sqrt{100u^2-64u^2}=\sqrt{36u^2}=6u.[/tex]
( u = units).
To find the length of CD, we will use the trigonometric function cosine:
[tex]\begin{gathered} cos36^{\circ}=\frac{AB}{DB}, \\ DB=\frac{10u}{cos36^{\circ}}\approx12.36u. \end{gathered}[/tex]
Finally, we get that:
[tex]CD=DB-BC=12.36u-8u=4.36u.[/tex]
Answer:
a)
[tex]\begin{equation*} \Delta ADB\approx\Delta CAB. \end{equation*}[/tex]
b)
[tex]\begin{gathered} BC=6\text{ units,} \\ CD\text{ }\approx4.36\text{ units.} \end{gathered}[/tex]
