Question

Help with this..but not Pauline helms she is unhelpful tutor.

Answer 1

Since the tangent of B is 4/3, then:

$\begin{gathered} \frac{AC}{BA}=\frac{4}{3} \\ \Rightarrow AC=\frac{4}{3}BA \end{gathered}$

On the other hand, since ABC is a right triangle with hypotenuse BC, then:

$(BA)^2+(AC)^2=(BC)^2$

Substitute AC=3/4 BA and BC=15 to find BA:

$\begin{gathered} (BA)^2+(\frac{4}{3}BA)^2=15^2 \\ \Rightarrow BA^2+\frac{16}{9}BA^2=15^2 \\ \Rightarrow(1+\frac{16}{9})BA^2=15^2 \\ \Rightarrow\frac{25}{9}BA^2=15^2 \\ \Rightarrow BA^2=\frac{9}{25}\cdot15^2 \\ \Rightarrow BA=\sqrt[]{(\frac{9}{25}\cdot15^2)} \\ \Rightarrow BA=\frac{3}{5}\cdot15 \\ \Rightarrow BA=9 \end{gathered}$

Substitute BA=12 into the expression for AC to find its value:

$AC=\frac{4}{3}BA=\frac{4}{3}(9)=12$

On the other hand, we know that:

$BD+DA=BA$

Substitute BA=12 and DA=3 to find BD:

$\begin{gathered} BD+3=9 \\ \Rightarrow BD=9-3 \\ \Rightarrow BD=6 \end{gathered}$

Finally, since BDE and BAC are similar triangles, we know that:

$\begin{gathered} \frac{DE}{BD}=\frac{AC}{BA} \\ \Rightarrow DE=\frac{AC}{BA}\times BD \end{gathered}$

Substitute AC=12, BA=9 and BD=6 to find the length DE:

$\begin{gathered} DE=\frac{12}{9}\times6 \\ \Rightarrow DE=8 \end{gathered}$

Therefore, the length of DE is:

$8$