Area of a Triangle (1 Viewer)

[cossine]

We need to find the area of the colored triangle (image is given below) with coordinates (0, 0), (20, 0), (20, 9). This will mean that the area of the triangle is 90 units^2 since 20*9/2 = 90. However if we follow the steps from b to e we will see the area of the triangle is 90.5.

(b) Show that the yellow (lower) right triangle has area 27.5.
(c) Show that the red rectangle has area 45.
(d) Show that the blue (upper) right triangle has area 18.
(e) Add the results of parts (b), (c), and (d), showing that the area of the colored
region is 90.5.

Question: Why do we get a different result?

 

[bruhmoment.]

Okay i think i got it, if you look at both the axis, the numbers aren't in a uniform manner for both axis, meaning for the x - axis, its increasing by 3, whereas for the y axis, its increasing by 2. so thats why you would get different results from the two ways you did. hope that helps

 

[Luukas.2]

Hint: The diagram is deliberately misleading.

It encourages the reader to make an assumption that is not justified.

Just because something appears to be true, doesn't mean it is true... what is the diagram implying is true that isn't actually known for sure from the information given?

Spoiler: Here are two possible options...

If the yellow and blue triangles and the red rectangle make up a triangle, then the vertex where all three meet cannot be at (11, 5)... in that case, all the calculations in (b), (c), and (d) are wrong.

Alternatively, if the vertex common to the yellow, blue, and red regions is (11, 5), then the three regions combine to form a quadrilateral rather than a triangle because the hypotenuses of the two triangles meet at at an angle that is not 180 degrees.

In other words:

• EITHER (0, 0) to (20, 9) is a straight line, in which case the rectangle's corner is at (11, 4.95) rather than at (11, 5) and the calculations should be:
  • yellow = 0.5 x 11 x 4.95 = 27.225 sq units
  • red = 9 x 4.95 = 44.55 sq units
  • blue = 0.5 x 9 x 4.05 = 18.225 sq units
  • Combined area of triangle = 27.225 + 44.55 + 18.225 = 90 sq units, as expected
• OR (0, 0) to (20, 9) to (11, 5) to (0, 0) forms a triangle of area 0.5, which is the extra area found by the above method as it is outside the area of the triangle from (0, 0) to (20, 0) to (20, 9) to (0, 0)

There is a third option, as well...

Spoiler: The third option...

The vertex where the yellow and blue triangles and the red rectangle meet if they do make up the larger triangle, could be at y = 5, rather than at x = 11 (as was the first case above). In that case, what appears to be the point (11, 5) will actually be at (100/9 , 5), and the calculations in (b), (c), and (d) are again wrong:

• yellow = 0.5 x 100/9 x 5 = 250/9 sq units
• red = (20 - 100/9) x 5 = 80/9 x 5 = 400/9 sq units
• blue = 0.5 x (20 - 100/9) x 4 = 0.5 x 80/9 x 4 = 160/9 sq units
• Combined area of triangle = 250/9 + 400/9 + 160/9 = (250 + 400 + 160)/9 = 810/9 = 90 sq units, as expected

 

[WeiWeiMan]

We need to find the area of the colored triangle (image is given below) with coordinates (0, 0), (20, 0), (20, 9). This will mean that the area of the triangle is 90 units^2 since 20*9/2 = 90. However if we follow the steps from b to e we will see the area of the triangle is 90.5.

(b) Show that the yellow (lower) right triangle has area 27.5.
(c) Show that the red rectangle has area 45.
(d) Show that the blue (upper) right triangle has area 18.
(e) Add the results of parts (b), (c), and (d), showing that the area of the colored
region is 90.5.

Question: Why do we get a different result?


View attachment 41997

The thing isn't a triangle so you can't just do the area like 1/2 bh

 

[cossine]

Hint: The diagram is deliberately misleading.

It encourages the reader to make an assumption that is not justified.

Just because something appears to be true, doesn't mean it is true... what is the diagram implying is true that isn't actually known for sure from the information given?

Spoiler: Here are two possible options...

If the yellow and blue triangles and the red rectangle make up a triangle, then the vertex where all three meet cannot be at (11, 5)... in that case, all the calculations in (b), (c), and (d) are wrong.

Alternatively, if the vertex common to the yellow, blue, and red regions is (11, 5), then the three regions combine to form a quadrilateral rather than a triangle because the hypotenuses of the two triangles meet at at an angle that is not 180 degrees.

In other words:

• EITHER (0, 0) to (20, 9) is a straight line, in which case the rectangle's corner is at (11, 4.95) rather than at (11, 5) and the calculations should be:
  • yellow = 0.5 x 11 x 4.95 = 27.225 sq units
  • red = 9 x 4.95 = 44.55 sq units
  • blue = 0.5 x 9 x 4.05 = 18.225 sq units
  • Combined area of triangle = 27.225 + 44.55 + 18.225 = 90 sq units, as expected
• OR (0, 0) to (20, 9) to (11, 5) to (0, 0) forms a triangle of area 0.5, which is the extra area found by the above method as it is outside the area of the triangle from (0, 0) to (20, 0) to (20, 9) to (0, 0)

There is a third option, as well...

Spoiler: The third option...

The vertex where the yellow and blue triangles and the red rectangle meet if they do make up the larger triangle, could be at y = 5, rather than at x = 11 (as was the first case above). In that case, what appears to be the point (11, 5) will actually be at (100/9 , 5), and the calculations in (b), (c), and (d) are again wrong:

• yellow = 0.5 x 100/9 x 5 = 250/9 sq units
• red = (20 - 100/9) x 5 = 80/9 x 5 = 400/9 sq units
• blue = 0.5 x (20 - 100/9) x 4 = 0.5 x 80/9 x 4 = 160/9 sq units
• Combined area of triangle = 250/9 + 400/9 + 160/9 = (250 + 400 + 160)/9 = 810/9 = 90 sq units, as expected

Thank you.