Answer as a fraction = 1/7
Answer in decimal form = 0.1428571
The decimal value is approximate.
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Explanation:
We'll be adding and subtracting areas, before we can divide the areas mentioned.
When I write something like "area(ACE)", I'm referring to the area of triangle ACE. It's to save time.
Any time there are 3 letters in the parenthesis for the area function, I'm talking about a triangle. Four letters refers to a quadrilateral. The order of the letters doesn't matter.
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Let's focus on triangle ACE. If we rotate it around so that AC is horizontal, then x is the base. Let h be the height. We can then say:
area(ACE) = 0.5*base*height = 0.5xh
since triangle AEG is the same area, we have
area(AEG) = 0.5xh
and furthermore,
area(ACG) = area(ACE) + area(AEG)
area(ACG) = 0.5xh+0.5xh
area(ACG) = xh
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Let's list out three facts:
- Segment AC is parallel to segment DB. By the alternate interior angle theorem, this means angle ACE = angle HDG
- Point G is the midpoint of segment CD, so we know CG = GD
- By the vertical angles theorem, angle AGC = angle HGD
In short,
- angle ACE = angle HDG
- segment CG = segment GD
- angle AGC = angle HGD
Those observations allow us to use the angle side angle (ASA) congruence theorem to prove that triangle ACG is congruent to triangle HDG. Statements 1 and 3 refer to the "angle" part of "ASA", while statement 2 is the side sandwiched in between said angles.
The key takeaway is that triangles ACG and HDG have the same area. Congruent triangles are identical clones of each other.
So,
area(HDG) = area(ACG)
area(HDG) = xh
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Now let's focus on triangle DEB
This triangle has side lengths 3 times larger compared to the smaller counterpart triangle ACE. These two triangles are similar (we can prove that by the Angle Angle Similarity Theorem)
Since the side lengths are 3 times larger, this makes the area 3^2 = 9 times larger
area(DEB) = 9*area(ACE)
area(DEB) = 9*(0.5xh)
area(DEB) = 4.5xh
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Next we'll use these two facts
- area(DEB) = 4.5xh
- area(HDG) = xh
to find that...
area(EGHB) = area(DEB) - area(HDG)
area(EGHB) = 4.5xh - xh
area(EGHB) = 3.5xh
Another way to find this area is to compute area(HAB) and subtract off the value of area(AEG). You should get the same result as above.
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Hopefully everything makes sense so far. If not, then review the earlier sections or feel free to ask about any step mentioned. There's a lot to keep track of.
At this point we have the areas we need.
- area(AEG) = 0.5xh
- area(EGHB) = 3.5xh
which I'll call m and n respectively
Divide m over n to get:
m/n = (0.5xh)/(3.5xh)
m/n = (0.5)/(3.5)
m/n = 5/35
m/n = 1/7 which is the exact fraction form
m/n = 0.1428571 which is the approximate decimal form
As you can see, the values of x and h don't matter because they cancel out at the end.
