Graph triangles

$\therefore$ the base of the red triangle is $\frac13$ of $6$ and the height is $\frac13$ of $12$

$\Rightarrow\text{P}$ has coordinates $(4,4)$

Check:

$-2\times4+12=4$ so it is on the line $y=-2x+12$

Using length and area scale factors

$y$-intercept: $x=0\Rightarrow y=12$

$x$-intercept: $y=0\Rightarrow 0=-2x+12\Rightarrow x=6$

The area scale factor from the red to the blue triangle is $4$

$\therefore$ the length scale factor is $\sqrt4=2$

Image

Base: total $6$ units

$6$ split in the ratio $2:1$ is $4:2$

$\Rightarrow x$ coordinate of $\text P$ is $4$.

Height: total $12$ units

$12$ split in the ratio $1:2$ is $4:8$

$\Rightarrow y$ coordinate of $\text P$ is $4$.

So $\text P$ is $(4,4)$.

Check:

$-2\times4+12=4$ so it is on the line $y=-2x+12$

Using the ratios between the sides of the triangles

$y$-intercept: $x=0\Rightarrow y=12$

$x$-intercept: $y=0\Rightarrow 0=-2x+12\Rightarrow x=6$

Image

$y=-2x+12$ has gradient $-2$ so heights are twice bases

$\therefore a+b=6$ (adding the bases),

$2a+2b=12$ (adding the heights),

$2a=-2b+12$ ($\text P$ is on the line $y=-2x+12$), but those equations tell us the same information

Areas: $\frac12\times b\times2b=4\times\frac12\times a\times2a$

$\therefore b^2=4a^2\Rightarrow b^2=(2a)^2$, so $b=\pm2a$.

$a$ and $b$ are both positive, so $b=2a$.

$a+b=6$ gives $a+2a=6\Rightarrow3a=6$ so $a=2$.

So $b=2a=4$, and point $\text P $ is $(b,2a)=(4,4)$.

Using more algebra

Image

$y$-intercept: $x=0\Rightarrow y=12$

$x$-intercept: $y=0\Rightarrow 0=-2x+12\Rightarrow x=6$

$\text P$ lies on $y=-2x+12$ so $b=-2a+12$.

The area of the blue triangle is $4$ times the area of the red triangle, so:

$\frac12a(12-b)=4\times\frac12(6-a)b$

Substitute $b=-2a+12$ into this:

$$\begin{align}&\tfrac12a\left(12-(-2a+12)\right)=4\times\tfrac12(6-a)(-2a+12)\\

\\

\Rightarrow & \tfrac12a(2a)=2\times(6-a)\left(2(6-a)\right)\\

\Rightarrow& a^2=\left(2\times(6-a)\right)^2\\

\Rightarrow& a=\pm2(6-a)\end{align}$$

$a$ and $6-a$ are both lengths on the graph so they are both positive so $a=2(6-a)$

$a=2(6-a)=\Rightarrow a=12-2a$ so $3a=12\Rightarrow a=4$

And $b=12-2a$, so $b=4$ too, and $\text P$ is $(4,4)$.