Question

find the lengths of the sides of the triangle . is it a right triangle? is it an isosceles triangle? P ( 3,−2,−3),R ( 7,0,1), Q(1,2,1)

Answer 1

• PR = PQ = 6
• QR = √40 ≈ 6.32
• isosceles triangle (not a right triangle)

Explanation:

You want the side lengths and the classification of the triangle with vertices P(3, -2, -3), R(7, 0, 1) and Q(1, 2, 1).

Distance formula

The lengths of the sides can be found using the distance formula in three dimensions:

d = √((x2 -x1)² +(y2 -y1)² +(z2 -z1)²)

This function is conveniently the "norm" of the vector with components (Δx, Δy, Δz).

The attachment shows the side lengths are ...

• PR = PQ = 6
• QR = √40 ≈ 6.32

Classification

Two of the side lengths are 6, so the triangle is isosceles. The long side is less than √2 times that, so the triangle is not a right triangle. It is an acute triangle.

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