Question

show that the points (0,7,10),(-1,6,6) and (-4,9,6) forms an isoceles right angled triangle.

Answer 1

Find the distance of the points
Two distance will be same
And the other one will be different
That is the reason it will be isosceles triangle

Hope it helps you

Answer 2

$\Large{\textbf{\underline{\underline{According\:to\:the\:Question}}}}$

Assumption

${\boxed{\sf\:{Points\;be\;PQR}}}$

Hence,

PQ = √{(-1 - 0)² + (6 - 7)² + (6 - 10)²}

PQ = √{(-1)² + (-1)² + (-4)²}

PQ = √(1 + 1 + 16)

PQ = √18

PQ = 3√2

$\textbf{\underline{Here\;we\;get:-}}$

PQ = 3√2

Now,

QR = √{(-4 + 1)² + (9 - 6)² + (6 - 6)²}

QR = √{(-3)² + (3)² + (0)²}

QR = √(9 + 9)

QR = √18

QR = 3√2

$\textbf{\underline{Here\;we\;get:-}}$

QR = 3√2

Now,

RP = √{(0 + 4)² + (7 - 9)² + (10 + 6)²}

RP = √{(4)² + (-2)² + (4)²}

RP = √(16 + 4 + 16)

RP = √(20 + 16)

RP = √36

RP = 6

$\textbf{\underline{Here\;we\;get:-}}$

RP = 6

Therefore,

PQ² + QR² = RP²

(3√2)² + (3√2)² = (6)²

18 + 18 = 36

36 = 36

By the Pythagoras theorem PQR is right angled triangle

$\Large{\boxed{\sf\:{We\;observed\;that\;points\;are\; vertices\;of\;right\; angled\;triangle}}}$