   Chapter 12.1, Problem 10E

Chapter
Section
Textbook Problem

# Find the lengths of the sides of the triangle PQR. Is it a right triangle? Is it an isosceles triangle?10. P(2, −1, 0), Q(4, 1, 1), R(4, −5, 4)

To determine

To find: The lengths of the sides of the triangle PQR and to verify whether the triangle PQR is a right triangle or not, an isosceles triangle or not.

Explanation

Consider three sides of the triangle (PQR) to be PQ, QR, and PR.

Formula:

Write the expression to find the distance between two points P1(x1,y1,z1) and P2(x2,y2,z2) .

|P1P2|=(x2x1)2+(y2y1)2+(z2z1)2 (1)

Here,

P1(x1,y1,z1) is the point, which is considered as one of the vertices of triangle PQR, and

P2(x2,y2,z2) is the point, which is considered as another vertex of triangle PQR.

Calculation of length of the side PQ:

In equation (1), substitute 2 for x1 , 1 for y1 , 0 for z1 , 4 for x2 , 1 for y2 , and 1 for z2 to find the length of the side PQ in the triangle PQR.

|PQ|=(42)2+[1(1)]2+(10)2=4+4+1=9=3

Thus, the length of the side PQ in the triangle PQR is 3.

Calculation of the length of the side QR:

In equation (1), substitute 4 for x1 , 1 for y1 , 1 for z1 , 4 for x2 , 5 for y2 , and 4 for z2 to find the length of the side QR in the triangle PQR

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