Question

Verify the following:
(i) (0, 7, –10), (1, 6, – 6) and (4, 9, – 6) are the vertices of an isosceles triangle.
(ii) (0, 7, 10), (–1, 6, 6) and (– 4, 9, 6) are the vertices of a right angled triangle.
(iii) (–1, 2, 1), (1, –2, 5), (4, –7, 8) and (2, –3, 4) are the vertices of a parallelogram.

Answer 1

(i) Let points (0, 7, -10), (1, 6, -6), and (4, 9, -6) be denoted by A, B, and C respectively
AB=\(\sqrt{(1-0)^2+(6-7)^2+(-6+10)^2}\)
= \(\sqrt{(1)^2+(-1)^2+(4)^2}\)
= \(\sqrt{1+1+16}\)
= \(\sqrt{18}\)
= 3\(\sqrt{2}\)
BA = \(\sqrt{(4-1)^2+(9-6)^2+(-6+6)^2}\)
=\(\sqrt{(3)^2+(3)^2}\)
= \(\sqrt{9+9}\)=\(\sqrt{18}\)=3\(\sqrt{2}\)
CA = \(\sqrt{(0-4)^2+(7-9)^2+(-10+6)^2}\)
= \(\sqrt{(-4)^2+(-2)^2+(-4)^2}\)
= \(\sqrt{16+4+16}\)=\(\sqrt{36}\)=6
Here, AB = BC ≠ CA
Thus, the given points are the vertices of an isosceles triangle.
(ii) Let (0, 7, 10), (-1, 6, 6), and (-4, 9, 6) be denoted by A, B, and C respectively.
AB=\(\sqrt{(-1-0)^2+(6-7)^2+(6-10)^2}\)
= \(\sqrt{(-1)^2+(-1)^2+(-4)^2}\)
= \(\sqrt{1+1+16}\)= \(\sqrt{18}\)
= 3√2
BC=\(\sqrt{(-4+1)^2+(9-6)^2+(6-6)^2}\)
= \(\sqrt{(-3)^2+(3)^2+(0)^2}\)
= \(\sqrt{9+9}\)= \(\sqrt{18}\)
= 3\(\sqrt2\)
CA = \(\sqrt{(0+4)^2+(7-9)^2+(10-6)^2}\)
=\(\sqrt{(4)^2+(-2)^2+(4)^2}\)
= \(\sqrt{16+4+16}\)
= \(\sqrt{36}\)
= 6
Now, AB²+BC² = \((3\sqrt2)^2+(3\sqrt2)^2\) = 18+18=36= AC²
Therefore, by Pythagoras theorem, ABC is a right triangle.
Hence, the given points are the vertices of a right-angled triangle.
(iii) Let (-1, 2, 1), (1, -2, 5), (4,-7, 8), and (2,-3, 4) be denoted by A, B, C, and D respectively.
AB=\(\sqrt{(1+1)^2+(-2-2)^2+(5-1)^2}\)
= \(\sqrt{4+16+16}\)
=\(\sqrt{36}\) = 6
BC= \(\sqrt{(4-1)^2+(-7+2)^2+(8-5)^2}\)
= \(\sqrt{9+25+9}=\sqrt{43}\)
CD=\(\sqrt{(2-4)^2+(-3+7)^2+(4-8)^2}\)
=\(\sqrt{4+16+16}\)
=\(\sqrt{36}\)
=6
DA = \(\sqrt{(-1-2)^2+(2+3)^2+(1-4)^2}\)
= \(\sqrt{9+25+9}=\sqrt{43}\)
Here, AB = CD = 6, BC = AD =\(\sqrt{43}\)
Hence, the opposite sides of quadrilateral ABCD, whose vertices are taken in order, are equal.
Therefore, ABCD is a parallelogram.
Hence, the given points are the vertices of a parallelogram.