1.consider the point a=(-1,0,1), b=(0,-2,3),and c=(-4,4,1) to be vertices of a triangle∆.evaluate all side lengths of ∆.
2.let ∆ be the triangle with vertices the points p=(3,1,-1),q=(2,0,3)and r=(1,1,1).determine whether ∆ is a right angle triangle.if it is not ,explain with reason,why?
1
Expert's answer
2021-06-23T15:13:00-0400
Solution.
1.a=(−1;0;1);b=(0;−2;3);c=(−4;4;1);
ab=(0+1)2+(−2−0)2+(3−1)2=3;
bc=(−4−0)2+(4+2)2+(1−3)2=214;
ac=(−4+1)2+(4−0)2+(1−1)2=5;
2.p=(3,1,−1),q=(2,0,3),r=(1,1,1);
pq=(2−3)2+(0−1)2+(3+1)2=32;
pr=(1−3)2+(1−1)2+(1+1)2=22;
qr=(1−2)2+(1−0)2+(1−3)2=6;
If the triangle is right-angled, then the square of the hypotenuse is equal to the sum of the squares of the legs:
pq2=pr2+qr2;
18=8+6=14, the triangle is not right-angled.
Answer: 1)3;214;5;
2) A triangle is not right-angled because Pythagoras' theorem does not hold.
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