Question

does the given set of points determine a plane? circle yes or no.
1)
yes no
2)
yes no
write whether the given points are collinear or not collinear.
1)
points a, b and c
2)
points x, y and z
choose whether the points on the plane are coplanar or not coplanar.
1)
a) not coplanar
b) coplanar
2)
a) not coplanar
b) coplanar

Explanation:

Step1: Determine if points form a plane

Three non - collinear points determine a plane. In the first part (1), points \(A\), \(B\), \(C\) are non - collinear, so they determine a plane (yes). In the second part (2) of the first question, three non - collinear points \(P\), \(Q\), \(R\) also determine a plane (yes).

Step2: Check collinearity

For collinearity, points lie on the same straight line. In the second question (1), points \(A\), \(B\), \(C\) lie on the same line (collinear). In (2), points \(X\) is not on the line with \(Y\) and \(Z\) (not collinear).

Step3: Check coplanarity

Coplanar points lie on the same plane. In the third question (1), \(S\), \(T\) are on the plane and \(U\) is not (not coplanar). In (2), \(B\), \(C\), \(D\) are on the same plane (coplanar).

Answer:

1. Yes (for the first set of points \(A\), \(B\), \(C\) in the first question), Yes (for the second set of points \(P\), \(Q\), \(R\) in the first question); 2) Collinear (for \(A\), \(B\), \(C\) in the second question), Not collinear (for \(X\), \(Y\), \(Z\) in the second question); 3) a) Not coplanar (for \(S\), \(T\), \(U\) in the third question), b) Coplanar (for \(B\), \(C\), \(D\) in the third question)