Exam

Olympiad workbook

Subject

IOQM

Chapter

Properties of Triangles

Olympiad workbook · IOQM · Properties of Triangles

In a triangle ABC , right-angled at A , the altitude through A and...

In a triangle ABC , right-angled at A , the altitude through A and the internal bisector of $\angle \mathrm{A}$ have lengths 3 and 4, respectively. Find the length of the median through A .

Step-by-step solution

$$\begin{aligned} & \angle \mathrm{CAE}=45^{\circ}=\angle \mathrm{BAE} \\ & \mathrm{AD}=3\end{aligned}$$ Let $B C=a, C A=b, A B=c$ $$\begin{aligned} & \frac{1}{2} b c=\frac{1}{2} \cdot a \cdot 3 \Rightarrow b c=3 a \\ & \frac{2 b c}{b+c} \cos \frac{A}{2}=4 \Rightarrow \frac{6 a}{b+c} \cdot \frac{1}{\sqrt{2}}=4 \\ & \Rightarrow 2 \sqrt{2}(b+c)=3 a \\ & \Rightarrow 8\left(b^2+c^2+2 b c\right)=9 a^2 \\ & \Rightarrow 8\left(a^2+6 a\right)=9 a^2 \\ & \Rightarrow 48 a=a^2 \Rightarrow a=48 \end{aligned}$$ So $\mathrm{AF}=\frac{\mathrm{a}}{2}=24$