Quote:
A triangle in the xy-coordinate plane has vertices with coordinates (7, 0), (0, 8), and (20, 10). What is the area of this triangle?
A. 72
B. 80
C. 87
D. 96
E. 100
acacia wrote:
JackReacher wrote:
Area of trapezoid formed by points (0,0) (8,0) (20,0) (20,10) = ((8+13)/2)*20 = 180
Area of Triangle 1 with points (7,0) (20,0) and (20,10) = 1/2 * 13*10 = 65
Area of Triangle 2 with points (0,0) (7,0) and (8,0) = 1/2 * 7*8 = 28
Required Area = Area of Trapezoid minus Area of Triangle 1 minus Area of triangle 2 = 180-65-28 = 87
I hope it's not confusing without an image.
I'm actually quite confused. These points (0,0) (8,0) (20,0) (20,10) are forming a triangle with points (0,0) (8,0) (20,0) in one line. And from definition, trapezoid is a quadrilateral with 2 parallel lines. Can you please explain again.
Attachment:
BOXMETHODTRAPEZOID.png [ 15.25 KiB | Viewed 27435 times ]
acacia , you are correct about those points. They make a straight line.
I think
JackReacher got the coordinates mixed up. Reverse (8,0) --> (0,8). The figure is indeed a trapezoid. Look at the figure above. Trapezoid area - green triangles' total area = original triangle.
JackReacher decided to use a trapezoid instead of a rectangle to form right triangles around the original triangle.
And I decided to draw the picture because for me this method of finding the area of an off-kilter triangle is much faster than the formula from hell.
The trapezoid is a clever variation on an easy way to solve this problem called the "Box Method," (which @mynhausen sketched in 2013 and I diagram below):
Typically, draw a rectangle around the triangle.
You find the area of the box you have drawn, and subtract the area of right-angled triangles that the box creates, leaving you with the area of the off-kilter triangle. Here that original triangle is ABC.
1.
Draw the smallest possible "box" around the triangle ABC. In the diagram, that is rectangle JKBM 2. Use the given vertices, and stay parallel to the axes.
List the coordinates of the rectangle's vertices. Here those coordinates are J (0,0), K (0,10), B (20, 10), M (20,0). You will have created a rectangular shape with right-angled triangles.
3. Find the area of the triangles that surround the original. In the diagram, those triangles are pastel yellow, blue, and purple.
To find the area of the right triangles, you need
the length of their legs. EASY to find. Segment JC, for example, has x-coordinates of 7 and 0. Voila, the length is 7.
4. The area of right triangles is easy to find, and is always (leg length * leg length * 1/2).
PURPLE triangle area is \(\frac{(7 * 8)}{2}\) = 28
YELLOW triangle area is \(\frac{(2 * 20)}{2}\) = 20
BLUE triangle area is \(\frac{(13*10)}{2}\) = 65
5. (PURPLE + YELLOW + BLUE) + WHITE = AREA OF RECTANGLE. Isolate the white triangle ABC.
Add the three triangles whose areas are known: 28 + 20 + 65 = 113
Area of rectangle is 20 * 10 = 200
200 - 113 = 87
6. Area of original triangle ABC is 87
Answer C
**In coordinate geometry, when you need a triangle's area but the triangle's sides are NOT parallel to the x- and y-axes, one method you can use is the box method because it creates right triangles whose areas are easy to find.
Hope it helps.
Attachment:
BOXmethod.png [ 11.59 KiB | Viewed 27067 times ]
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