The scale of a map is 5 cm : 1 km. A reservoir has an actual area of 0.7 km^2. Find the area, in square centimetres, of the reservoir on the map. Answer ... cm^2 ABCD is a rhombus. X and Y are points on BD such that DX = YB. Show that triangle ADX is congruent to triangle ABY. Give a reason for each statement you make. Answer

1. Convert the linear map scale into an area scale.

Scale: $5\text{ cm} : 1\text{ km}$

So,
$1\text{ km}^2 \text{ is represented by } 5^2\text{ cm}^2 = 25\text{ cm}^2.$

2. Find the map area corresponding to the actual area of $0.7\text{ km}^2$.

$0.7\text{ km}^2 \text{ is represented by } 0.7 \times 25 = 17.5\text{ cm}^2.$

1. State a pair of equal sides from the rhombus.

Since $ABCD$ is a rhombus,

$AD = AB$

(all sides of a rhombus are equal).

2. State the given pair of equal segments on the diagonal.

Given that

$DX = YB.$

3. Show the included angles are equal.

Since $ABCD$ is a rhombus, opposite sides are parallel, so

$AD \parallel BC \quad \text{and} \quad AB \parallel DC.$

As $X$ and $Y$ lie on $BD$, the angles $\angle ADX$ and $\angle ABY$ are the angles made by the same diagonal $BD$ with the parallel sides $AD$ and $AB$ in the rhombus. Hence

$\angle ADX = \angle ABY.$

4. Conclude triangle congruence with a valid congruence test.

Therefore, in $\triangle ADX$ and $\triangle ABY$,

$AD = AB, \quad DX = YB, \quad \angle ADX = \angle ABY.$

Hence

$\triangle ADX \cong \triangle ABY$

by SAS.