Find ratio of area of p to area of q.

- cwrw238

Find ratio of area of p to area of q.

- schrodinger

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- cwrw238

|dw:1341591452831:dw|

- eyust707

well they are both just triangles minus the same little corner...

- cwrw238

The whole figure is a square
and the line bisects the right side of the square.
this question was from a maths paper for 16 year old's.
I would have struggled to do this at 16 - probably would have given up.

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- eyust707

|dw:1341591736170:dw|
p = (1*2)/2 - A
q= (2*2)/2 - A
Subtracting the same amount from both wont effect the ratio, so:
p:q = 1/2

- eyust707

*i think*** lol i havent taken geometry since 10th grade

- eyust707

haha thats a cool one though!

- eyust707

What do you think does my logic seem sound?

- cwrw238

no - thats not the answer

- eyust707

hmm maybe you can tjust take off the A

- eyust707

but if we find that little A we are golden

- cwrw238

yes

- eyust707

I could do it using trig but thats a pain and I know there's an easier way...

- cwrw238

i did it differently using systems of equations
there maybe an easier way though
this is how i started

- cwrw238

|dw:1341592406267:dw|

- phi

|dw:1341592415019:dw|
we can label the sides of the square 1 for easy calculation.
similar triangles p and x, with x scaled by 1/2 means the area of x is 1/4 of p
the rest of the areas are easy to find.

- cwrw238

area x : area y = 1^2 : 2^2 = 1: 4
then i got 4 equations in x, y , p and q
and splved
- a bit messy !!

- cwrw238

yea - i think thats a bit better way than mine

- eyust707

haha phi isnt that just the same problem again... |dw:1341592696668:dw|

- phi

the altitudes are also in ratio of 1/2 so p's alt is 1/3 and x's is 1/6
as they add to 1/2

- cwrw238

yes

- phi

so find the area of p,
area of q is (1/2)^2 + (1/2)*(1/2)^2 + area(x)

- cwrw238

so q = 1/4 + 1/8 + area(x) = 3/8 + area(x)

- phi

area(x)= (1/2)*(1/2)*(1/6)= 1/24
area(p)= (1/2)*1*(1/3)= 1/6

- phi

so ratio of areas p/q = 2/5

- cwrw238

i got 2: 5
area p = 1/6
area q = 3/8 + 1/24 = 10/24 = 5/12
p/q = 1/6 * 12/5 = 12/60 = 2/5
- yea

- cwrw238

- quite a searching question for 16 year old, i thought

- cwrw238

- thanx

- phi

Back in the old days, maybe they worked harder at this stuff??

- cwrw238

yea - good point!

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