# Thread: Ratio of areas of similar figures

1. ## Ratio of areas of similar figures

Hi all,
Here goes the question:

In the diagram, AC,RQ and BP are parallel lines. AR=9cm,RQ=6cm,BP=8cm, and RB=3cm. Calculate the ratio of:
(i)area of triangle ARQ:area of trapezium RBPQ'
(ii)area of triangle BQR:area of triangle ARQ
(iii)area of triangle ABC:area of triangle BQR

Another question:

In the diagram, AB=5cm,BE=4cm,EC=2cm,CD=3cm and DA=1cm. If DE=xcm, using a geometrical argument, deduce the value of x.

2. In Q2 are the two triangles similar?

3. nope it isn't stated in the question but anyway it doesn't look similar

4. please can someone help me because i need it for my assisgnment. thanks again.

5. We're not supposed to do work that contributes to a student's final grade, so you'll need to show us what you've tried...

6. you got me wrong, by saying assignment i am referring to normal homework. I mean i have no idea how to start it off so can anyone provide me with hints?

7. Hello, AeroScizor!

$\displaystyle \text{In the diagram: }\:AC \parallel RQ \parallel BP.$

$\displaystyle AR=9,\; RQ=6,\;BP=8,\;RB=3.$

$\displaystyle \text{Calculate the ratios of the areas of:}$

$\displaystyle (i)\;\Delta ARQ : \text{quad }RBPQ$

We have: .$\displaystyle \Delta ARQ \sim \Delta ABP$

Hence: .$\displaystyle AR:AB \:=\:9:12 \;=\;3:4$

Then: .$\displaystyle \text{area }\Delta ARQ:\text{area }\Delta ABP \;=\;9:16$

Therefore: .$\displaystyle \Delta ARQ:\text{quad }RBPQ \;=\;9:7$

$\displaystyle (ii)\;\Delta BQR : \Delta ARQ$

$\displaystyle \Delta BQR$ has a base of 3.
$\displaystyle \Delta ARQ$ has a base of 9.
. . Their bases are in the ratio $\displaystyle 3:9 \:=\:1:3$

They have the same height.

Therefore, their areas are in the ratio $\displaystyle 1:3.$

$\displaystyle (iii)\; \Delta ABC : \Delta BQR$

We have: .$\displaystyle \Delta ABC \sim \Delta BQR$

Their bases are in the ratio $\displaystyle 12:3 \:=\:4:1$
Their heights are also in the ratio $\displaystyle 4:1$

Therefore, their areas are in the rato $\displaystyle 16:1$

8. thanks a million. i really wanted to know the thought process