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# Geometry Question

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I have been stuck on this problem for a bit. I understand the basic steps, but I can't seem to get the area for some reason! Here's a picture of my problem(instructions added below the picture):

I need to find the perimeter and area of  △XYZ. The coordinates are as follows: X(0,1), Y(4,-4), and Z(-5,-2). If I'm correct, the perimeter is approximately 21.45. I found this because, using the distance formula, XZ ≈ 21.45, XY ≈ 6.4, and YZ ≈ 9.22, which, added together equals 21.45(could someone verify?).

But what I really need help with is area. I have already done some of these kinds of problems, but this one is really getting me somehow. I can't figure it out! Any help would be great. NOTE: ROUND THE AREA TO THE NEAREST TENTH(they only wanted perimeter in hundreths).

JBDBIBBaerman  Mar 10, 2017

#2
+7266
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I know this has already been answered but here's another way of looking at it:

To find the area, you can find the area of the big rectancle and subtract the areas of the green, red, and orange triangles. Then you're left with just the area of the blue triangle.

area = rectangle - green - red - orange

area = (5)(9) - (1/2)(5)(3) - (1/2)(4)(5) - (1/2)(9)(2)

area = 45 - 7.5 - 10 - 9

area = 18.5 square units

hectictar  Mar 10, 2017
#1
+88839
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Perimeter

XZ = sqrt ( 5^2 + 3^2)  = sqrt (34)

XY = sqrt ( 4^2 + 5^2)  = sqrt (41)

YZ = sqrt ( 9^2 + 2^2)  = sqrt (85)

sqrt (34) + sqrt(41) + sqrt (85) ≈  21.45  units.....correct  !!!

The area is a little difficult.....there are several ways to do this using trig.....but....I'm going to assume you might not have had that.....so....let's do this

The slope between Y(4,-4), and Z(-5,-2)  is (-2 - -4) / (-5 -4)  = 2/-9  = -2/9

And the equation of the line joining  these two points  is   y = (-2/9)(x - 4) - 4  →

y = (-2/9x + 8/9 - 4 →  y = (-2/9)x - 28/9

So...... a perpendicular line from the apex of the triangle - point X - to this line will serve as the height of the triangle.....and the slope of this perpendicular line will be the negative reciprocal of -2/9   = 9/2

And the equation of this line will be

y = (9/2) ( x - 0) + 1   → y =  (9/2)x + 1

Setting these lines equal, we can find the x coordinate of their intersection

(-2/9)x - 28/9   =  (9/2)x + 1

-28/9 - 1  =  (9/2 + 2/9)x

-37/9   = (85/18)x     multiply both sides by 18/85

(-37/9)(18/85)  =  -74/85

And the y coordinate of their intersection can be found  as  (9/2)(-74/85) + 1  = -248/85

So the distance from X -(0,1)- to  (-74/85, -248/85)  is given by

sqrt [ (74/85)^2 + ( 1 + 248/85)^2 ]  = 37/ sqrt (85)  = the triangle's  height

So.....the area of a triangle is (1/2) B * H

The Base is YZ   = sqrt (85)....so the area =

(1/2)sqrt(85) * 37 / sqrt (85)   -  (1/2)(37)  = 18.5 square units  !!!

Whew!!!!....that was a lot of work......but....I verified the answer here :

http://www.wolframalpha.com/input/?i=area+triangle+%5B+(0,1),+(4,-4),++(-5,-2)%5D

Hope that helps !!!

CPhill  Mar 10, 2017
#6
+139
+5

It did a lot.

JBDBIBBaerman  Mar 13, 2017
#2
+7266
+11

I know this has already been answered but here's another way of looking at it:

To find the area, you can find the area of the big rectancle and subtract the areas of the green, red, and orange triangles. Then you're left with just the area of the blue triangle.

area = rectangle - green - red - orange

area = (5)(9) - (1/2)(5)(3) - (1/2)(4)(5) - (1/2)(9)(2)

area = 45 - 7.5 - 10 - 9

area = 18.5 square units

hectictar  Mar 10, 2017
#5
+139
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Thanks for another way of doing it! The more ways there are, the more ways there are to verify an answer.

JBDBIBBaerman  Mar 13, 2017
#3
+88839
+5

Very nice, hectictar.....I like your method MUCH better....!!!

CPhill  Mar 10, 2017
#4
+139
+5

Thank you both so much! I see where I made some mistakes now. Thank you!

JBDBIBBaerman  Mar 13, 2017