The triangle whose vertices are (0,7,-10), (1,6,-6) and (4,9,-6) is a ___. Options: A . &nbsp acute-angled triangle B . &nbsp Right-angled triangle C . &nbsp scalene triangle D . &nbsp isosceles triangle

Answer: Option B : B and D Let the vertices of the triangle be A(0,7,-10), B(1,6,-6) and C(4,9,-6) A B = √ ( 1 − 0 ) 2 + ( 6 − 7 ) 2 + ( − 6 − ( − 10 ) ) 2 = √ 1 2 + 1 2 + 4 2 = √ 18 B C = √ ( 4 − 1 ) 2 + ( 9 − 6 ) 2 + ( − 6 − ( − 6 ) ) 2 = √ 3 2 + 3 2 + 0 2 = √ 18 A C = √ ( 4 − 0 ) 2 + ( 9 − 7 ) 2 + ( − 6 − ( − 10 ) ) 2 = √ 4 2 + 2 2 + 4 2 = √ 36 H e r e , A B = B C & A B 2 + B C 2 = A C 2 Hence, the triangle is isosceles as well as right-angled.

The triangle whose vertices are (0,7,-10), (1,6,-6) and (4,9,-6) is a __

Question

The triangle whose vertices are (0,7,-10), (1,6,-6) and (4,9,-6) is a ___.

Options:

A . acute-angled triangle

B . Right-angled triangle

C . scalene triangle

D . isosceles triangle

Answer: Option B
:
B and D

Let the vertices of the triangle be A(0,7,-10), B(1,6,-6) and C(4,9,-6)
$A B = \sqrt{(1 - 0)^{2} + (6 - 7)^{2} + (- 6 - (- 10))^{2}} = \sqrt{1^{2} + 1^{2} + 4^{2}} = \sqrt{18} B C = \sqrt{(4 - 1)^{2} + (9 - 6)^{2} + (- 6 - (- 6))^{2}} = \sqrt{3^{2} + 3^{2} + 0^{2}} = \sqrt{18} A C = \sqrt{(4 - 0)^{2} + (9 - 7)^{2} + (- 6 - (- 10))^{2}} = \sqrt{4^{2} + 2^{2} + 4^{2}} = \sqrt{36}$
$H e r e, A B = B C & A B^{2} + B C^{2} = A C^{2}$
Hence, the triangle is isosceles as well as right-angled.