See attachment. If a line Y=A cuts the triangle in the

Question

See attachment. If a line Y=A cuts the triangle in the figure such that it creates 2 equal halfs by area what is A?

Mishari · Answer

Area of triangle = 1/2 x base x hight = 1/2 x 6 x 4 = 12

Since the line Y = A divides the triangles into two different shapes with same area, each region will have area of 1/2 x 12 = 6

Let's look at the top region, the new smaller triangle formed. Let's call its base M [part of the line Y=A] and its hight N.

MN/2 = 6 --> MxN=12

divide the bottom region into a rectangle and a triangle
The area of the rectangle = (4-N) x M = 4M-NM
The area of the triangle = 1/2 x (6-M) x (4-N) = (24-6N-4M+MN)/2
Area of triangle + area of rectangle = 4M-NM+12-3N-2M+(MN/2)
= 2M-3N+12-(NM/2) = 6 = 2M-3N-(NM/2) = -6 --> 4M-12N-NM =-12
So,
4M-12N-12=-12--> 4M - 12N = 0 --> M = 3N
MxN=12 --> 3N^2=12 --> N^2=4 --> N = 2

A = 4 - N = 4 - 2 = 2 = A

Answer: A = 2

I'm sure I've either made a mistake or there is a much simpler and easier solution or even both :-D

vshaunak@gmail.com · Answer

The area of the big triangle = 1/2 *4*6 = 12
As the y=A cuts the triangle into two halves, the area of the traingle above with base line y =A will be 6

Equation of the line joining (0,4) and (6,0) is:
x/6 + y/4 = 1
putting the values y =A will give the co-ordinate of the intersection of the y =a with the hypotenuse of the big triangle.
point of intersection = ( /2, A)

the area of the triangle above with base line y=a is 
1/2 * ( /2 * (4-A)=6
solving this equation A =4- sqrt(8)

javed · Answer

Can you explain how you came up the the points of intersection ( /2,A). 

Javed.

Cheers!

ywilfred · Answer

Just equate area of triangle with area of trapezium.

cicerone · Answer

Dear friend there's a big mistake with your solution..........
As you said if N=2 then M must be 6 because M=3N.
But from the diagram it is clear that M must be less than 6 because the base of original triangle is 6.

So i think the answer is not 2.

cicerone · Answer

This is correct but at the end the answer will not be 2.
Solving this we get A= 4-sqrt(8)

This must be the answer.

There is a simpler way to answer this if you are comfortable with Similar Triangles concept.

The original triangle and smaller triangle will be similar to each other
The ratio of areas of similar triangles will be equal to the squares of their corresponding sides....
i.e. ABC is similar to ADE

Ratio of areas = AB^2/ AD^2 = 2/1
Since AB^2 = 16 AD^2 = 8
i.e AD must be sqrt(8)

So A = AB-AD = 4-sqrt(8)

Juaz · Answer

getting a = 4-2sqrt2

from similar triangle 
4/6=(4-a)/b => b=(12-3a)/2

As area of top triangle is half of total traingle area
=> 1/2*(1/2*4*6) = 1/2*(4-a)(12-3a)/2 
=> 24 = (a-4)*(3a-12)
=> a2-8a+8 = 0
possible a = 4-2sqrt2

vshaunak@gmail.com · Answer

Yes, you are right. A should not be 2. It should be 4-sqrt(8). I did not solve it correctly.

jayt696969 · Answer

Answer is A = 3

Area of triangle is 1/2BH = 1/2*6*4 = 12

If you cut a line Y=A into the triangle and split its area in half, then the upper piece, which is a smaller triangle will have an area of 6.

1/2*B*H = 6 

B*H = 12

The factors of 12 or 1,2,3,4,6,12 and the only one that fits is Y=3 by process of elimination. If Y were 1, then X would be 12, but thats outside the original triangle, etc....So the answer is Y=A=3

jayt696969 · Answer

i retract my answer because the point (4,3) is also outside of the triangle.

KillerSquirrel · Answer

I think that y = a = 1

 :-D

See attachment. If a line Y=A cuts the triangle in the

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