Verify the following:

Question 7 of 13intermediate🔧 ApplyLong Answer6 marks

(i)

$(0, 7, -10)$ , $(1, 6, -6)$ and $(4, 9, -6)$ are the vertices of an isosceles triangle.

Answer

The points $(0, 7, -10)$ , $(1, 6, -6)$ and $(4, 9, -6)$ form an isosceles triangle with $AB = BC = 3\sqrt{2}$ units.

(ii)

$(0, 7, 10)$ , $(-1, 6, 6)$ and $(-4, 9, 6)$ are the vertices of a right angled triangle.

Answer

The points (0, 7, 10), (-1, 6, 6) and (-4, 9, 6) form a right-angled triangle at vertex B.

(iii)

$(-1, 2, 1)$ , $(1, -2, 5)$ , $(4, -7, 8)$ and $(2, -3, 4)$ are the vertices of a parallelogram.

Answer

The points $(-1, 2, 1)$ , $(1, -2, 5)$ , $(4, -7, 8)$ and $(2, -3, 4)$ form a parallelogram as opposite sides are equal.

Explanation

This question tests the application of the 3D distance formula to verify geometric properties. For isosceles triangles, two sides must be equal. For right-angled triangles, Pythagoras theorem must be satisfied. For parallelograms, opposite sides must be equal.

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