Question 10
$X$ is a point inside an equilateral triangle $ABC$. $Y$ is the foot of the perpendicular from $X$ to $AC$, $Z$ is the foot of the perpendicular from $X$ to $AB$, and $W$ is the foot of the perpendicular from $X$ to $BC$.
The ratio of the distances of $X$ from the three sides of the triangle is $1:2:4$ as shown in the diagram.
If the area of $AZXY$ is $13\text{ cm}^2$, find the area of $ABC$. Justify your answer. [5 marks]
Investigation (source; 2 bonus marks): If $XY:XZ:XW=a:b:c$, find the ratio of the areas of $AZXY$ and $ABC$.
Written reasoning is required. The reference answer is not an automatic assessment of a student proof.
The ratio of the distances of $X$ from the three sides of the triangle is $1:2:4$ as shown in the diagram.
If the area of $AZXY$ is $13\text{ cm}^2$, find the area of $ABC$. Justify your answer. [5 marks]
Investigation (source; 2 bonus marks): If $XY:XZ:XW=a:b:c$, find the ratio of the areas of $AZXY$ and $ABC$.
Written reasoning is required. The reference answer is not an automatic assessment of a student proof.