Problem 3 · The perimeter with the altitude drawn
From the area to the base, then Pythagoras on both triangles.
Integer answer, at most 4 digitsTriangle $ABC$ in the figure has area 84 m². The altitude $AD$ has length 8 m. If side $BC$ is divided into 7 equal parts, point $D$ is the second division point. What is the triangle's perimeter?
Copa Cangur · SCM
Medium
Closed answer
Reasoned solution
Key idea: the area gives the base; the position of $D$ splits the base into two known legs.
$$84 = \frac{BC \cdot 8}{2} \;\Longrightarrow\; BC = 21\ \text{m}.$$
Each seventh is $3$ m; $D$ is the second point: $BD = 6$ and $DC = 15$. Pythagoras on triangles $ABD$ and $ADC$:
$$AB = \sqrt{8^{2}+6^{2}} = 10, \qquad AC = \sqrt{8^{2}+15^{2}} = 17.$$
$$P = 21 + 10 + 17 = 48\ \text{m}.$$
Answer: 48