Question

points c, d, and g lie on plane x. points e and f lie on plane y. which statements are true? select from options. there are exactly two planes that contain points a, b, and f. there is exactly one plane that contains points e, f, and b. the line that can be drawn through points c and g would lie in plane x. the line that can be drawn through points e and f would lie in plane y. the only points that can lie on plane y are points e and f.

Explanation:

Step1: Analyze the first statement

Three non - collinear points determine a plane. Points \(A\), \(B\), and \(E\) are non - collinear. There is exactly one plane that contains three non - collinear points. So, the statement “There are exactly two planes that contain points \(A\), \(B\), and \(E\)” is false.

Step2: Analyze the second statement

Points \(E\), \(F\), and \(B\) are non - collinear. Three non - collinear points determine a plane. So, the statement “There is exactly one plane that contains points \(E\), \(F\), and \(B\)” is true.

Step3: Analyze the third statement

Points \(C\) and \(G\) lie on plane \(X\). A line passing through two points on a plane lies on that plane. So, the statement “The line that can be drawn through points \(C\) and \(G\) would be in plane \(X\)” is true.

Step4: Analyze the fourth statement

Points \(E\) and \(F\) lie on plane \(Y\). A line passing through two points on a plane lies on that plane. So, the statement “The line that can be drawn through points \(E\) and \(F\) would lie in plane \(Y\)” is true.

Step5: Analyze the fifth statement

Plane \(Y\) contains points \(E\) and \(F\), but there can be infinitely many points on plane \(Y\). So, the statement “The only points that can lie on plane \(Y\) are points \(E\) and \(F\)” is false.

Answer:

• There is exactly one plane that contains points \(E\), \(F\), and \(B\).
• The line that can be drawn through points \(C\) and \(G\) would be in plane \(X\).
• The line that can be drawn through points \(E\) and \(F\) would lie in plane \(Y\).