Question

(a) If the areas of two similar triangles are equal, prove that they are congruent.

(b) The scale of the map is 1 : 200000. A plot of land with area 20 $k{m}^2$ is to be represented on the map. Find:

(i) The number of kilometres on the ground which is represented by 1 cm on the map.
(ii) The area in sq. km that can be represented by $1c{m}^2$
(iii) The area of the map that represents the plot of land. [6 MARKS]

Answer 1

Each part: 3 Marks

(a) Let $A_1,A_2$ be the areas of the triangles and $S_2,S_2$ and $S_3$ be their corresponding sides

$\begin{array}{ccc}& \frac{A_1}{A_2}-\frac{s_1^2}{s_1^2}=\frac{s_2^2}{s_2^2}=\frac{s_3^2}{s_3^2}& [where{s}_1,s_2,s_{3}representsthesidesoftriangles]\\ But& A_2=A_2& \\ \therefore & \frac{S_1^2}{S_1^2}=\frac{s_2^2}{s_2^2}=\frac{s_3^2}{s_3^2}=1\Rightarrow s_1=s_2,s_2=s_{2}and{s}_3=s_3& \end{array}$
$\therefore$ Triangles are congruent (SSS congruence rule)


(b) Scale factor,

$k=\frac{1}{200000}and1cm=\frac{1}{200000}\times$ Length of the ground
(i) Length of the ground $=\frac{200000}{100\times 1000}km=2km$
(ii) Area of the map =$k^2$ times the corresponding area of the ground
$1c{m}^2={\left(\frac{1}{200000}\right)}^2\times$ Area of ground
$\Rightarrow$ Area of ground $=200000\times 200000c{m}^2=4k{m}^2$
(iii) Total area of the map = $k^2$ times the total area of the plot
$={\left(\frac{1}{200000}\right)}^2\times 20k{m}^2=\frac{1}{40000000000}\times 20\times (100\times 1000{)}^{2}c{m}^2=5c{m}^2$