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ajsz033
Posts: **169**Registered User Junior Member

In the xy-coordinate plane, three vertices of a triangle are (1,2), (7,5), and (7,9). What is the area of the figure?

This is a grid in question. I had to use the distance formula to find the distance of all three sides of the triangle and then use Heron's formula to find the area.

S=(1/2)(A+B+C)

A=[S(S-A)(S-B)(S-C)]^(1/2)

This problem took me about 5 minutes because my method is so tedious and I easily make errors. Can anyone tell me if there is a shortcut to the problem?

This is a grid in question. I had to use the distance formula to find the distance of all three sides of the triangle and then use Heron's formula to find the area.

S=(1/2)(A+B+C)

A=[S(S-A)(S-B)(S-C)]^(1/2)

This problem took me about 5 minutes because my method is so tedious and I easily make errors. Can anyone tell me if there is a shortcut to the problem?

Post edited by ajsz033 on

## Replies to: Find area: Given 3 coordinates of a triangles

1,026Registered User Senior Member359Registered User MemberTook about a minute to do using my method. Basically you should sketch out the triangle with points and coordinates, but do not really follow any scale, just mark coordinates. Then use the following formula: S=1/2 base*height to find area of triangle. Also, you will have to introduce another point (7,2) and first find the area of the triangle with points (1,2),(7,2),(7,9). The area of this large triangle is S=1/2*(7-1)*(9-2)=21. Now find the area of triangle with points (1,2),(7,5),(7,2) using the same method. S=1/2*(7-1)*(5-2)=9. The answer to the original question is the latter subtracted from the former, i.e., 21-9= 12. Is this the correct answer?

3,447Registered User Senior MemberPlot the three coordinates just to get a visual of what you're solving for. The base of the triangle is easy to find since (7, 9) is just 4 vertical units above (7, 5). The height is the distance from (1, 2) to the line x = 7 (imagine a dotted line going from (1, 2) to (7,2) ). Therefore, height = 7 -1 = 6.

Area = 1/2 (base)(height)

= 1/2 (4)(6)

= 12

359Registered User Member92Registered User Junior Member63Registered User Junior Member359Registered User Member582Registered User Member1,520- Senior Member33Registered User Junior MemberFind the vector going from A to B, we'll call it V

Find the vector going from A to B, we'll call it U

Find the cross product of V and U

Find the length of the cross product and divide by 2 and DONE!

yeah seems like a long process but it can be done really quick on a calculator

582Registered User Member1,316Registered User Senior MemberSo we know that the area of a triangle is base x height/2. Notice that two of the points on the triangle have a common x-coordinate (7). So we can use these coordinates to form the base. The length of the base is 9 - 5 = 4, since that is the difference between the y-coordinates of the two points.

The next thing is to find the height, which is simply 7 minus the x co-ordinate of the third point - i.e. 7 - 1 = 6.

Now we do 6 * 4/2 = 12.

It might be worth imagining the triangle in your head or doing a rough sketch. Also note that I kind of turned the graph around to form the base and height of the triangle.