There has got to be an easier way...?

Question

is there some type of formula to figure this problem out?

Find the area of a triangle with the vertices A(0, 2) B(8,2) and C(4, -3)

Also a parallelogram is given, and I would like to know the formula or if there is a program out there for the TI-89 so I could do this easier....because I do not want to use the distance formula and draw in an altitude(apothem), I think it would be sweet if I could just plug in the vertices and it will find the area :)

This deals with graphing points on a graph paper.

The parallelogram vertices are

A(-4, 2) B(1, 6) C(15, 6) and D(10, 2)
I am given no degree measures whatsoever in this problem.

Answer 1

There is an easier way for these problems, because one side or another is aligned with an axis. This makes it very easy to calculate the area without any difficult equations.

In your triangle, A (0,2) and B (8,2) are on a horizontal line at height y=2. That makes it very easy to (1) compute the distance from A to B (it's the difference in their x-values), and (2) compute the perpendicular distance to C (the differences between y=2 and C's y-value).

The formula for a triangle is 1/2 base * height.

Using AB as the base, the length of the base is 8 (8 - 0 = 8). The altitide to C from y=2 is 5 (2 - (-3) = 5).

So this particular triangle's area is:

1/2 * base * height
1/2 * 8 * 5
20

===================

The parallelogram problem is similarly easy. The points you supplied are:

A(-4, 2) B(1, 6) C(15, 6) and D(10, 2)

Note that AD is parallel to the X axis (at y=2), and so is BC (at y=6).

So... the base of your parallelogram is 14 (15 - 1, the difference in x-values of B and C), and the height is 4 (distance from y=2 to y=6).

The area of a parallelogram is:

base * height =
14 * 4 =
56

That's exactly the same answer one of the respondents below came up with, but does not involve a complicated equation, dividing it up into two triangles, etc.

Answer 2

Instead of telling you that you dont need the formula or solving the example in some way or another (the answer to which I'm sure you already have), I will actually answer your question.

If a triangle has vertices:
(a,b) , (c,d) , (e,f), the area is:

A = | [(a-e)(d-f) - (b-f)(c-e)] / 2 |
(absolute value of the expression is taken)

for a parallelogram, or any quadrilateral for that matter, you could now break it up into two triangles by drawing a diagonal and find the area of each triangle. (Actually, if you know it is a parallelogram this is unnecessary, as the diagonals of a parallelogram divide it evenly. I go through it anyway to show the technique.)

So, in your parallelogram,
A(-4, 2) B(1, 6) C(15, 6) and D(10, 2)

There are two triangles

1)
A(-4, 2) B(1, 6) C(15, 6)
choose one point whose values are subtracted from the other two points. I'll choose (1,6). The other points become:
(-4-1, 2-6) = (-5, -4) and (15-1, 6-6) = (14, 0)

Area = | [(-5)(0) - (-4)(14)] /2 |
= 28


2)
A(-4, 2) C(15, 6) D(10, 2)
Choose (10,2) to subtract from the others. The points become:
(-4-10, 2-2) = (-14,0)
(15-10, 6-2) = (5, 4)
Area = | [((-14)(4) - (0)(5)] / 2 |
=
28

Total = 28 + 28 = 56

OR, you could just use the following:
▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼

THE GENERAL FORMULA for the area of a quadrilateral with verticies: (x1, y1) , (x2, y2) , (x3, y3) , (x4, y4) [where the verticies are numbered clockwise or counter-clockwise around the quadrilateral] is:

Area =
| [(x2-x1)(y3-y1) -(x3-x1)(y2-y1) +(x3-x1)(y4-y1) -(x4-x1)(y3-y1)] / 2 |

▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲

| [(1-(-4))(6-2) - (15-(-4))(6-2) + (15-(-4))(2-2) - (10-(-4))(6- 2)] /2 |

= | [ 5*4 - 19*4 + 19*0 - 14*4 ]/2 |
= | [20 - 76 + 0 - 56]/2 |
= | -112/2 | = | -56 |
= 56

Answer 3

Area of a triangle = 1/2 base * height

base is 8 units long as you can see from the distance from A to B. Then the height from here is 5. You get this by subtracting the y component of C from A or B.

In general it's a bit more complex to find the area from just three co ordinates. Essentially what you want to do is just find the length of a line between two points (the base), and then find the perpendicular distance from this line to your third point (the height), and then A= 1/2 base*height.


The formula for the area of a parallelogram is
length of base * vertical height * sin(smaller angle inside the parallelogram)

Answer 4

From A to B, we know that the base is 8. Height is [2 - (-3)] = 5.
Area of triangle = 1/2 X 8 X 5 = 20 square unit.

Answer 5

Join the points and make a big triangle, then substract the little triangle

(11*2)/2-(3*2)/2 = 11-3=9

Answer is 9

Answer 6

I'm sure there are programs you can use, but its not really hard to find the hypotenuse of one tryangle for the height and use base * height/2