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## cwrw238 3 years ago Find ratio of area of p to area of q.

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1. cwrw238

|dw:1341591452831:dw|

2. eyust707

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

3. 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.

4. 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

5. eyust707

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

6. eyust707

haha thats a cool one though!

7. eyust707

What do you think does my logic seem sound?

8. cwrw238

no - thats not the answer

9. eyust707

hmm maybe you can tjust take off the A

10. eyust707

but if we find that little A we are golden

11. cwrw238

yes

12. eyust707

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

13. cwrw238

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

14. cwrw238

|dw:1341592406267:dw|

15. 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.

16. 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 !!

17. cwrw238

yea - i think thats a bit better way than mine

18. eyust707

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

19. 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

20. cwrw238

yes

21. phi

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

22. cwrw238

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

23. phi

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

24. phi

so ratio of areas p/q = 2/5

25. 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

26. cwrw238

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

27. cwrw238

- thanx

28. phi

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

29. cwrw238

yea - good point!

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