Square intersected by a line in ratio 1:5

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MathsFormula

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I could not find a diagram to insert here so I'll try to describe the image the best I can.

A sloping line divides the area of a square in the ratio 1:5.

The line intersects the square from one corner and then meets the other side and the distance is marked "b". The other side of the line on the square is marked "a".
"a" appears to be longer than "b".
The intersection forms a triangle with a side "b" and a trapezium with side "a".

Question asks " What is the ratio a:b"

My calculation:

Let the area of the triangle be 1. So using the area of a triangle formula: AREA = 0.5*base*height

So 1 = 0.5*b*(a+b)

Area of trapezium = 0.5 (a+b)*height

So 5 = 0.5 (a+b)(a+b)

I could not solve these to find a value for "a" or "b"?

Thanks
 
MathsFormula said:
I could not find a diagram to insert here so I'll try to describe the image the best I can.

A sloping line divides the area of a square in the ratio 1:5.

The line intersects the square from one corner and then meets the other side and the distance is marked "b". The other side of the line on the square is marked "a".
"a" appears to be longer than "b".
The intersection forms a triangle with a side "b" and a trapezium with side "a".

Question asks " What is the ratio a:b"

My calculation:

Let the area of the triangle be 1. So using the area of a triangle formula: AREA = 0.5*base*height

So 1 = 0.5*b*(a+b)

Area of trapezium = 0.5 (a+b)*height

So 5 = 0.5 (a+b)(a+b)

I could not solve these to find a value for "a" or "b"? You do not NEED to find a value for "a" or "b"

Thanks
Click to expand...

Area of the rectangle = (a + b)2

Area of triangle = 1/2 * b * (a + b)

(Area of triangle)/(Area of the rectangle) = 1/6 = [1/2 * b * (a + b)]/[(a + b)2]

Now calculate a:b from above
 
Subhotosh Khan said:
Area of the rectangle = (a + b)2

Area of triangle = 1/2 * b * (a + b)

(Area of triangle)/(Area of the rectangle) = 1/6 = [1/2 * b * (a + b)]/[(a + b)2]

Now calculate a:b from above
Click to expand...

Carrying on from above:

1/6=0.5b/(a+b)

a+b=3b

a = 2b

So ratio a:b is 1:2 FANTASTIC. Thank you

Please note that the answer in the book is 2:1 so perhaps I've made an error in my calculations or have not described the shape properly.

I need to probe your mind please Subhotosh. I need to know what made you think of calculating the ratio of 1:6 for the triangle and square and not 1:5 triangle:trapezium.
If I know what makes you think that way I may improve my maths.


This bit posted later:

I've tried your method using the 1:5 ratio and it works out the same too. I made an error in my first post (above) in my calculation of the area of a trapezium.

Area of trapezium is:
0.5 (a+(a+b))(a+b)

Area of triangle is : 0.5b (a+b)

So 1/5 = (0.5b (a+b))/(0.5 (a+(a+b)(a+b))


1/5 = b /(a+(a+b)

a+(a+b) =5b

2a+b = 5b

2a = 4b

a = 2b

1:2

Same answer Subhotosh but I would still like to know why you decided to do the calculation using 1/6. Thanks
 
Last edited:
Just realised that "a" is meant to be longer than "b" so answer should indeed be 2:1 and not 1:2
 
MathsFormula said:
I need to probe your mind please Subhotosh. I need to know what made you think of calculating the ratio of 1:6 for the triangle and square and not 1:5 triangle:trapezium.
If I know what makes you think that way I may improve my maths.
Click to expand...

Area of the square = (Area of the triangle) + (Area of the trapezium)

(Area of the triangle) / (Area of the trapezium) = 1/5 → (Area of the triangle) = A & (Area of the trapezium) = 5*A →

Area of the square = (Area of the triangle) + (Area of the trapezium) = A + 5*A = 6*A →

(Area of the square) / (Area of the triangle) = 6*A/A = 6/1
 
Subhotosh Khan said:
Area of the square = (Area of the triangle) + (Area of the trapezium)

(Area of the triangle) / (Area of the trapezium) = 1/5 → (Area of the triangle) = A & (Area of the trapezium) = 5*A →

Area of the square = (Area of the triangle) + (Area of the trapezium) = A + 5*A = 6*A →

(Area of the square) / (Area of the triangle) = 6*A/A = 6/1
Click to expand...

I'm stuck on the second line. I don't know what you mean.



Can trouble you for one more answer please?

Why do you think I got the answer 1:2 and not 2:1 as in the book?
 
Subhotosh Khan said:
Area of the square = (Area of the triangle) + (Area of the trapezium)

(Area of the triangle) / (Area of the trapezium) = 1/5 → (Area of the triangle) = A & (Area of the trapezium) = 5*A →

Area of the square = (Area of the triangle) + (Area of the trapezium) = A + 5*A = 6*A →

(Area of the square) / (Area of the triangle) = 6*A/A = 6/1
Click to expand...

What do you mean by 'A'. What is this abbreviating?
 
We know a=2b

I said this meant 1:2 THIS IS WRONG

What I should have said is from a=2b
we get a/b = 2/1

This is the same as saying a:b=2:1

Voila

Is that the correct way of thinking?
 
Subhotosh, thanks for your help. Got there in the end.


Denis said:
So we can get rid of you: YES :ğrolleyes:
Click to expand...

Dennis (spelt the correct way) that was so hurtful :D

FYI Den2is I'm going to take over on the forum. Going to be the highest THREAD OPENER.
 
Don't fret Den2is. No more maths for me for a while I think. I'm sleeping and waking up thinking of equations. That's not good. May be back in a few weeks.

Thanks for your help
 
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