Coordinate Geometry

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Wed Dec 17, 2008 9:17 am

A triangle has vertices at (0,2), (4,7) and (6,0).
What is the area of the triangle?

(A) 19
(B) 20
(C) 21
(D) 22
(E) 23

Answer: A

Brent Hanneson - Creator of GMATPrepNow.com

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Source: Beat The GMAT — Problem Solving | Wed Dec 17, 2008 9:17 am

amitabhprasad: Master | Next Rank: 500 Posts; Posts: 259; Joined: Thu Jan 18, 2007 8:30 pm; Thanked: 16 times

by amitabhprasad » Wed Dec 17, 2008 9:29 am

Area of Rectangle = 7*6
Minus areas of 3 triangle =
1/2(6*2)=6
+1/2(2*7)=7+
+1/2(4*5) = 10
==> 42-23 = 19

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vittalgmat: Legendary Member; Posts: 621; Joined: Wed Apr 09, 2008 7:13 pmFollowed by: 4 members; Thanked: 33 times

by vittalgmat » Wed Dec 17, 2008 10:54 am

The formula for the area of triangle given the 3 coordinates is:

1/2 Abs [ (x1-x2)(y2-y3) - (y1-y2)(x2-x3) ]

Here (x1, y1), (x2,y2) and (x3,y3) are (0,2), (4,7) and (6,0).
(however anyone can be (x1,y1) etc. )
Substituting and simplifying,
we get 0.5 * |-28 -10|

=38/2 = 19

HT helps

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parallel_chase: Legendary Member; Posts: 1153; Joined: Wed Jun 20, 2007 6:21 amFollowed by: 2 members; Thanked: 146 times

by parallel_chase » Wed Dec 17, 2008 10:57 am

area of the triangle in a coordinate system

1/2 l (x2-x1)(y3-y2) + (y2-y1)(x3-x2) l

1/2 l(4-0)(7-0) + (7-2)(6-4)l

1/2 l28 + 10l

1/2 * 38 = 19

Hence A.

No rest for the Wicked....

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Wed Dec 17, 2008 3:52 pm

The formula for the area of triangle given the 3 coordinates is:

1/2 Abs [ (x1-x2)(y2-y3) - (y1-y2)(x2-x3) ]

Wow, I can't say that I've seen that formula before.
Very interesting.

Brent Hanneson - Creator of GMATPrepNow.com

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mrsmarthi: Senior | Next Rank: 100 Posts; Posts: 45; Joined: Sun Nov 30, 2008 8:41 pm; Thanked: 4 times

by mrsmarthi » Wed Dec 17, 2008 9:00 pm

My ans will be A - 19. But the approach to the solution is different.

With the the vertices of the triangle, I noticed that one point is on the X axis (6,0) and one on y axis (0,2) and the other point in 1 Quandrant.

I tried to construct a Trapeziod with the points A(0,0), B(6,0), C(4,7)(these are two of the vertices of the trianle in question)and D(0,7). Here CD is parallel to AB, AB and CD are the bases and AD is the height of the trapeziod. Now area of the trapoziod is 1/2(sum of the bases) * height = 1/2(6+4) * 7 = 35.

Let the E(0,2) be the third vertex which lies on AD. I tried finding the area of traingle AEB and area of triangle ECD which are the right triangles.

Area of triangle AEB = 1/2 * 6 * 2 = 6
Area of Triangle ECD = 1/2 * 4 * 5 = 10.

So Area of traingle BCE(which is our question) is Area of trapeziod ABCD - area of traingle AEB and area of triangle ECD.

35-10-6 = 19.

So my Ans is 19.

My apologies for not showing in a diagram. I don't know how can I put that in the diagram format. But if you try to draw the figure, you should be able to understand the points.

Brent Hanneson,

Thank you for posting some good questions. When you said that you are not aware of the formula that other posters have made, I realised there should be a way. And here is my way. Taking the inspiration of Thinking Out of the Box. [/img]

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Wed Dec 17, 2008 10:15 pm

I like this technique, mrsmarthi.
Rather than find the area of the rectangle and subtract the areas of the three triangles around the target triangle, you found the area of the trapezoid and subtracted only 2 triangles.

That's what I love about GMAT questions: 1 question and we already have 3 distinct solutions.

Any more out there?

Brent Hanneson - Creator of GMATPrepNow.com

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logitech: Legendary Member; Posts: 2134; Joined: Mon Oct 20, 2008 11:26 pmFollowed by: 25 members; Thanked: 237 times; GMAT Score:730

Re: Coordinate Geometry

by logitech » Wed Dec 17, 2008 11:14 pm

Brent Hanneson wrote:A triangle has vertices at (0,2), (4,7) and (6,0).
What is the area of the triangle?

(A) 19
(B) 20
(C) 21
(D) 22
(E) 23

Answer: A

Okay here is the life safer method for those who are not good at math. Put these three points on your paper ( It has grids ) Draw your triangle and count the squares!

You will be surprised!

LGTCH
---------------------
"DON'T LET ANYONE STEAL YOUR DREAM!"

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