Asked by Bob
A quadrilateral has vertices at (0,1), (3,4), (4,3) and (3,0). Its perimeter can be expressed in the form asqrt(2)+bsqrt(10) with a and b integers. What is the sum of a and b?
All Answers 1
Answered by
GPT-5 mini
AI
Compute the side lengths in order:
- (0,1) to (3,4): distance = sqrt((3-0)^2+(4-1)^2) = sqrt(9+9)=3√2
- (3,4) to (4,3): distance = sqrt(1^2+(-1)^2)=√2
- (4,3) to (3,0): distance = sqrt((-1)^2+(-3)^2)=√10
- (3,0) to (0,1): distance = sqrt((-3)^2+1^2)=√10
Perimeter = 3√2+√2+√10+√10 = 4√2+2√10, so a=4, b=2 and a+b=6.
- (0,1) to (3,4): distance = sqrt((3-0)^2+(4-1)^2) = sqrt(9+9)=3√2
- (3,4) to (4,3): distance = sqrt(1^2+(-1)^2)=√2
- (4,3) to (3,0): distance = sqrt((-1)^2+(-3)^2)=√10
- (3,0) to (0,1): distance = sqrt((-3)^2+1^2)=√10
Perimeter = 3√2+√2+√10+√10 = 4√2+2√10, so a=4, b=2 and a+b=6.
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