Question

How to find the area of the red triangle given in the pic?

Step by step answer with proof and verification.

Answer 1

✔️It turns out the problem can be solved with a simple calculation. The area of the red triangle is:

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✅79 + 10 – 72 – 8 = 9
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✔️The area of a triangle is (base × height)/2, and the area of a parallelogram is (base × height).

✔️A triangle whose base equals one side of the parallelogram, and whose height reaches the opposite side of the parallelogram, has exactly half the area of a parallelogram.

✔️The same is true for a pair of triangles, if the pair of triangles span one side and if their heights reach the opposite side.

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✔️Let’s first label the unknown areas with letters a, b, c, d, e, and f. And let’s label the area we want with the letter x.
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✔️First consider the two triangles below whose sides together span the top side (length) of the parallelogram.

✔️As both of these triangles touch the bottom side of the parallelogram, each triangle has the same vertical height as the parallelogram does between its bottom and top sides.

✔️Therefore, these two triangles have a combined area equal to 1/2 the area of the parallelogram. Their area is:

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✔️Upper triangles => (x + a) + (72 + b + 8)
=> (area of parallelogram)/2
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✔️Now consider the triangle with a base of the left side (width) of the parallelogram that touches the right width of the parallelogram.

✔️As this triangle touches two opposite sides of the parallelogram, the triangle has the same horizontal distance the parallelogram does between its left and right sides.

✔️Therefore, this triangle also has an area equal 1/2 the area of the parallelogram, and its area equals:

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✔️Left triangle => a + 79 + b + 10
=> (area of parallelogram)/2
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✔️As both equations equal half the area of the parallelogram, we can set these areas equal to each other.

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(x + a) + (72 + b + 8) = a + 79 + b + 10
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✔️We can cancel the terms a and b on both sides and then solve for x.

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x + 72 + 8 = 79 + 10

x= 79 + 10 – 72 – 8 = 9
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ANSWER: 9✔️

Answer 2

First consider the two triangles whose sides together span the bottom side (length) of the parallelogram. As both of these triangles touch the top side of the parallelogram, each triangle has the same vertical height as the parallelogram does between its bottom and top sides.



Therefore, these two triangles have a combined area equal to 1/2 the area of the parallelogram. Their area is:

Lower triangles = (c + 79 + e) + (d + 10 + f) = (area of parallelogram)/2

Now consider the triangles whose sides span the right side of the parallelogram and touch the left side of the parallelogram. As these triangles touch two opposite sides of the parallelogram, each triangle has the same horizontal distance the parallelogram does between its left and right sides.



Therefore, these triangles also have an area equal 1/2 the area of the parallelogram, and their combined areas equal:

Right triangles = (x + c + 72 + d) + (e+ 8 + f) = (area of parallelogram)/2

As both equations equal half the area of the parallelogram, we can set these areas equal to each other.

(c + 79 + e) + (d + 10 + f) = (x + c + 72 + d) + (e + 8 + f)

We can cancel the terms c, e, d, and f on both sides and then solve for x.

79 + 10 = x + 72 + 8

x = 79 + 10 – 72 – 8 = 9