Answer to Question #277168 in Analytic Geometry for Jovy

Question #277168

1. Solve for the area in square units of the right triangle with vertices F(0,7),U(2,-3),N(7,-2)




2. Find the area in square units of a rectangle whose vertices are L(0,6),O(2,-2),V(-2,-3),E(-4,5)






1
Expert's answer
2021-12-09T03:21:59-0500

1.

UF=02,7(3)=2,10)\overrightarrow{UF}=\langle0-2, 7-(-3)\rangle=\langle-2,10)\rangle

UN=72,2(3)=5,1)\overrightarrow{UN}=\langle7-2, -2-(-3)\rangle=\langle5,1)\rangle

UFUN=2(5)+10(1)=0=>UFUN\overrightarrow{UF}\cdot\overrightarrow{UN}=-2(5)+10(1)=0=>\overrightarrow{UF}\perp\overrightarrow{UN}

The right triangle FNUFNU


UF=(2)2+(10)2=226|\overrightarrow{UF}|=\sqrt{(-2)^2+(10)^2}=2\sqrt{26}

UN=(5)2+(1)2=26|\overrightarrow{UN}|=\sqrt{(5)^2+(1)^2}=\sqrt{26}

S=12UFUN=12(226)(26)=26(units2)S=\dfrac{1}{2}|\overrightarrow{UF}||\overrightarrow{UN}|=\dfrac{1}{2}(2\sqrt{26})(\sqrt{26})=26({units}^2)

2.


VO=2(2),2(3)=4,1)\overrightarrow{VO}=\langle2-(-2), -2-(-3)\rangle=\langle4,1)\rangle

VE=4(2),5(3)=2,8)\overrightarrow{VE}=\langle-4-(-2), 5-(-3)\rangle=\langle-2,8)\rangle

EL=0(4),65=4,1)=VO\overrightarrow{EL}=\langle0-(-4), 6-5\rangle=\langle4,1)\rangle=\overrightarrow{VO}

OL=02,6(2)=2,8)=VE\overrightarrow{OL}=\langle0-2, 6-(-2)\rangle=\langle-2,8)\rangle=\overrightarrow{VE}




VEVO=2(4)+8(1)=0=>VEVO\overrightarrow{VE}\cdot\overrightarrow{VO}=-2(4)+8(1)=0=>\overrightarrow{VE}\perp\overrightarrow{VO}



The rectangle VELOVELO


VE=(2)2+(8)2=217|\overrightarrow{VE}|=\sqrt{(-2)^2+(8)^2}=2\sqrt{17}

VO=(4)2+(1)2=17|\overrightarrow{VO}|=\sqrt{(4)^2+(1)^2}=\sqrt{17}

S=VEVO=217(17)=34(units2)S=|\overrightarrow{VE}||\overrightarrow{VO}|=2\sqrt{17}(\sqrt{17})=34({units}^2)


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