Question

Calculate the area of the shaded part of the triangle.

Answer 1

a)

Since DE // AC, then

Triangles BDE and BAC are similar

Then the ratio between their areas is the square of the ratio between their corresponding side

Since BD = 2 cm

Since DA = 4 cm, then

BA = 2 + 4 = 6 cm

The side BD in triangle BDE = 2

The side BA in triangle BAC = 6

The ratio between them is

`\begin{gathered} (BD)/(BA)=(2)/(6) \\ (BD)/(BA)=((2)/(2))/((6)/(2)) \\ (BD)/(BA)=(1)/(3) \end{gathered}`

We will use this ratio to find the ratio between their areas

`(A_(BDE))/(A_(BAC))=((BD)/(BA))^2`

Since the area of the triangle, BDE is 2 cm^2, then

`\begin{gathered} (2)/(A_(BAC))=((1)/(3))^2 \\ (2)/(A_(BAC))=(1)/(9) \end{gathered}`

By using the cross-multiplication

`\begin{gathered} A_(BAC)*1=2*9 \\ A_(BAC)=18cm^2 \end{gathered}`

To find area DECA subtract area triangle BDE from area triangle BAC

`\begin{gathered} A_(DECA)=18-2 \\ A_(DECA)=16cm^2 \end{gathered}`

The missing is 16