Question

(a) A vertical stick 12 cm long casts a shadow 8 cm long on the ground. At the same time, a tower casts a shadow 40 m long on the ground. Determine the height of the tower.

(b) The perimeter of two similar triangles are 30 cm and 20 cm, respectively, If one side of the first triangle is 12 cm, determine the corresponding side of the second triangle. [6 MARKS]

Answer 1

(a) Axiom: 1 Mark
Solution: 1 Mark
Correct answer: 1 Mark

(b) Axiom: 1 Mark
Formula: 1 Mark
Correct answer: 1 Mark

(a)


$\begin{array}{ccc}& \angle B=\angle Q& \left[Angleofelevationatthesametime\right]\\ and& \angle C=\angle R& \left[Each{90}^{\circ }\right]\\ \Rightarrow & \Delta ABC\sim \Delta PQR& \left(AASimilarity\right)\\ Thus,& \frac{AB}{PQ}=\frac{BC}{QR}=\frac{AC}{PR}& \\ & \frac{8}{4000}=\frac{12}{x}\Rightarrow x=6000cm& \\ \Rightarrow & x=60m& \end{array}$
$\therefore$ Height the tower = 60 m


(b) Suppose $\Delta ABC$ and $\Delta DEF$ are similar.
$\begin{array}{cc}Then,& \frac{AB}{DD}=\frac{BC}{EF}=\frac{AC}{DF}=\frac{AB+BC+AC}{DE+EF+DF}\\ & =\frac{Perimeterof\Delta ABC}{Perimeterof\Delta DEF}\\ \Rightarrow & \frac{P\left(\Delta ABC\right)}{P\left(\Delta DEF\right)}=\frac{A}{DE}\Rightarrow \frac{30}{20}=\frac{12}{DE}\\ \Rightarrow & DE=8cm\end{array}$