Another triangulation

Recall that the areas of triangles with equal altitudes are proportional to the bases of the triangles (see Euclid’s Elements of Geometry: Book 6 Proposition1). We have

$\dfrac{EA}{EB}=\dfrac{A_{CEA}}{A_{CEB}}=\dfrac{A_{FEA}}{A_{FEB}}$

$\dfrac{a+b+7}{3+7}=\dfrac{a}{3}$

$3a+3b+21=10a$

$3b-7a=-21$ $\color{#D61F06}(1)$

$\dfrac{DA}{DC}=\dfrac{A_{BDA}}{A_{BDC}}=\dfrac{A_{FDA}}{A_{FDC}}$

$\dfrac{a+b+3}{7+7}=\dfrac{b}{7}$

$7a+7b+21=14b$

$-7b+7a=-21$ $\color{#D61F06}(2)$

From $\color{#D61F06}(1)$ and $\color{#D61F06}(2)$ , we get $a=7.5$ and $b=10.5$ .

The area of the original triangle is $3+7+7+10.5+7.5=\boxed{35}$

I THINK THAT THIS IS INTERESTING!! I think that this might be thought by some as a bit of a cheat but I got the right answer with the method, so it must be alright to some extent! It took me a while to realize that these three areas completely determines the area of the big triangle. But thinking about it, the triangle to the left is 3 seventh of the middle triangle and so the left side of the triangle is determined. Similarly the right side of the triangle is determined, so the area of the triangle is completely determined! | So now why not do the question for a PARTICULAR triangle. This one has base length 7 and the height of the middle triangle is 2 (which gives the area of the middle triangle as 7, which is what we want! Note that this particular triangle has a right angle, because the right triangle has area 7. So the height of that triangle is 4. | As is often the case this is tricky to explain without a diagram! But I'll try (explaining the method, but not fully). The length of the line from the high point of the left triangle (with area 3) to the base of the whole triangle is 20/7 (because the area of the 'base triangle' is 7 whilst the area of the two triangles (one the base and one on the left) is 10. The length from this point on the base to the very right point is 5, because of congruent triangles, and so the length of the same point from the very left point is 2. So the height H of the large triangle is such that H/7 = (20/7)/2. So H = 10. And this triangle has a right angle as I said before. So its area is Base x Height / 2 = 7 x 10 / 2 = 35. Well I did get the correct answer, whether of not you approve of the method! Regards, David

By ladder theorem, let A be the area of the unshaded region of the triangle. Then,

1/10 + 1/14 = 1/7 + 1/(3+7+7+A)

Since we are solving for the area of the given triangle, then 3+7+7+A= 35.