Determine the area of the shaded region.
Solution 1 by Peter Denton, East Grand Rapids High School
First, place the regions in the xy-plane. Then the top point of the
shaded region (region 1) will be the intersection point P of the lines Now solve x + 1 = -2/5 x + 4 to get x = 15/7. So the height h of region 3 is : h = 5 - x
= 5 - 15/7 = 20/7. Therefore, the area of region 1 is 25/2 - 5/2 - 40/7 = 30/7. |
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Solution 2 by Ben Montgomery, Marquette High School
First, I set up the object in the xy-plane. Then I found the intersection point P of the two lines y = x + 1 and y = -2/5 x + 4. That gave me the third point of the triangle, ( 15/7 , 22/7 ). With all 3 points I used the distance formula Next, I simply used Heron’s formula for the area of a triangle:
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Also finding exact solutions were: Andrew Jeanguenat (De La Salle
Collegiate H.S.); Scott Schrimscher (Detroit Catholic Central H.S.); Catherine
Nezich (Marquette H.S.); Kiran Sebastian (Saginaw Heritage H.S.); Eva Dou (Grosse
Pointe South H.S.); Kim Lebioda (Plymouth H.S.); and Sara Ross (Plymouth Christian
Academy).
Those finding approximate solutions were: Alex Best, Zach Cresswell, Aaron Tennant (St. Louis H.S.); Kyle Grooms, Mandy Hoffmann, Lauren Kelley (Marshall H.S.); Kellie DeSchutter (Troy Athens H.S.); Trevor Lambert, Jake Maloney (Detroit Catholic Central H.S.); Shaun Pezeshki (West Bloomfield H.S.); Derek Banagis, Mischa Dylewski, Rachel Fear, Jason Jou, Alex Piskin, Drew Reyelts, Lindsey Westerhof (Forest Hills Eastern H.S.); Alex Mead, Angela Verkade (Forest Hills Northern H.S.); and Aaron Lorincz (Plymouth Christian Academy).
to find the 3 side lengths and came up with 
. 
where a,
b, and c are the side lengths and s = (a
+ b + c) / 2 is the semi-perimeter. Using 
and 
, I
found the area of the shaded region to be 30/7 square units.