Points, lines, and planes

randRange(-100, 100) / 150 randRange(-100, 100) / 500 randFromArray(["A", "F", "K", "U"]) randFromArray(["B", "G", "L", "V"]) randFromArray(["C", "H", "M", "W"]) randFromArray(["D", "I", "N", "X"]) randFromArray(["E", "J", "O", "Y"]) randFromArray(["R", "S", "T", "P"])

init({ range: [ [-1, 15], [-5, 8] ], scale: [ 30, 30 ] }); // plane path([ [6 + (0 - 6) * cos(TILT) - (0 - 2) * sin(TILT), 2 + (0 - 6) * sin(TILT) + (0 - 2) * cos(TILT)], [6 + (10 - 6) * cos(TILT) - (0 - 2) * sin(TILT), 2 + (10 - 6) * sin(TILT) + (0 - 2) * cos(TILT)], [6 + (12 - 6) * cos(TILT) - (4 - 2) * sin(TILT), 2 + (12 - 6) * sin(TILT) + (4 - 2) * cos(TILT)], [6 + (2 - 6) * cos(TILT) - (4 - 2) * sin(TILT), 2 + (2 - 6) * sin(TILT) + (4 - 2) * cos(TILT)], true ]); style({ arrows: "->" }, function() { // line in the plane line([6 + (3 - 6)* cos(TILT+TILT2) - (3 - 2)* sin(TILT+TILT2), 2 + ( 3 - 6 ) * sin(TILT+TILT2) + ( 3 - 2 ) * cos(TILT+TILT2)], [6 + (9 - 6)* cos(TILT+TILT2) - (1 - 2)* sin(TILT+TILT2), 2 + ( 9 - 6 ) * sin(TILT+TILT2) + ( 1 - 2 ) * cos(TILT+TILT2)]); // line in the plane (opposite direction hack to get double-headed arrow) line([6 + (9 - 6)* cos(TILT+TILT2) - (1 - 2)* sin(TILT+TILT2), 2 + ( 9 - 6 ) * sin(TILT+TILT2) + ( 1 - 2 ) * cos(TILT+TILT2)], [6 + (3 - 6)* cos(TILT+TILT2) - (3 - 2)* sin(TILT+TILT2), 2 + ( 3 - 6 ) * sin(TILT+TILT2) + ( 3 - 2 ) * cos(TILT+TILT2)]); // perpendicular line (top) line([6 + (6 - 6)* cos(TILT) - (2 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( 2 - 2 ) * cos(TILT)], [6 + (6 - 6)* cos(TILT) - (6 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( 6 - 2 ) * cos(TILT)]); // Perpendicular line (bottom) line([6 + (6 - 6)* cos(TILT) - (0 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( 0 - 2 ) * cos(TILT)], [6 + (6 - 6)* cos(TILT) - (-3 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( -3 - 2 ) * cos(TILT)]); }); // dashed line behind the plane style({ strokeDasharray: "- " }, function() { line([6 + (6 - 6)* cos(TILT) - (2 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( 2 - 2 ) * cos(TILT)], [6 + (6 - 6)* cos(TILT) - (0 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( 0 - 2 ) * cos(TILT)]); }); var r = randRange(0, 100) / 150; var theta = randRange(-314, 314) / 100; // points style({ r: 0.15, stroke: "none" }, function() { circle([6 + (4 - 6)* cos(TILT+TILT2) - (8/3 - 2)* sin(TILT+TILT2), 2 + ( 4 - 6 ) * sin(TILT+TILT2) + ( 8/3 - 2 ) * cos(TILT+TILT2)], { fill: PINK }); circle([6, 2], { fill: BLUE }); circle([6 + (8 - 6)* cos(TILT+TILT2) - (4/3 - 2)* sin(TILT+TILT2), 2 + ( 8 - 6 ) * sin(TILT+TILT2) + ( 4/3 - 2 ) * cos(TILT+TILT2)], { fill: GREEN }); circle([6 + (6 - 6)* cos(TILT) - (-2 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( -2 - 2 ) * cos(TILT)], { fill: ORANGE }); circle([6 + (17/6 + r*cos(theta) - 6)* cos(TILT) - (1.4 + r*sin(theta) - 2)* sin(TILT), 2 + ( 17/6 + r*cos(theta) - 6 ) * sin(TILT) + ( 1.4 + r*sin(theta) - 2 ) * cos(TILT)], { fill: "purple" }); }); label([6 + (4 - 6)* cos(TILT+TILT2) - (8/3 - 2)* sin(TILT+TILT2), 2 + ( 4 - 6 ) * sin(TILT+TILT2) + ( 8/3 - 2 ) * cos(TILT+TILT2)], A, "above right" ); label([6.1, 2], B, "above right"); label([6 + (8 - 6)* cos(TILT+TILT2) - (4/3 - 2)* sin(TILT+TILT2), 2 + ( 8 - 6 ) * sin(TILT+TILT2) + ( 4/3 - 2 ) * cos(TILT+TILT2)], C, "above right" ); label([6 + (6 - 6)* cos(TILT) - (-2 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( -2 - 2 ) * cos(TILT)], D, "right" ); label([6 + (17/6 + r*cos(theta) - 6)* cos(TILT) - (1.4 + r*sin(theta) - 2)* sin(TILT), 2 + ( 17/6 + r*cos(theta) - 6 ) * sin(TILT) + ( 1.4 + r*sin(theta) - 2 ) * cos(TILT)], E, "below" ); label([6 + (9.2 - 6)* cos(TILT+TILT2) - (1 - 2)* sin(TILT+TILT2), 2 + ( 9.2 - 6 ) * sin(TILT+TILT2) + ( 1 - 2 ) * cos(TILT+TILT2)], "\\ell", "right"); label([6 + (12.1 - 6)* cos(TILT) - (4 - 2)* sin(TILT), 2 + ( 12.1 - 6 ) * sin(TILT) + ( 4 - 2 ) * cos(TILT)], "\\mathcal{" + R + "}", "right");

What is another way to name plane \mathcal{R}?

A + B + E

A + C + E

A + E + B

A + E + C

B + A + E

B + C + E

B + E + A

B + E + C

C + A + E

C + B + E

C + E + A

C + E + B

E + A + B

E + A + C

E + B + A

E + B + C

E + C + A

E + C + B

Plane

Planes can be named with three noncollinear points.

Noncollinear points are points that are not on the same line.

Find any three points in the plane \mathcal{R} that are not on the same line and list them in any order.

For example, we can write \mathcal{R} as plane A + B + E, plane A + C + E, or plane B + E + C.

randFromArray([ "\\overleftrightarrow{" + A + B + "}", "\\overleftrightarrow{" + A + C + "}", "\\overleftrightarrow{" + B + A + "}", "\\overleftrightarrow{" + B + C + "}", "\\overleftrightarrow{" + C + A + "}", "\\overleftrightarrow{" + C + B + "}" ]) randFromArray([ "\\overleftrightarrow{" + A + "}", "\\overleftrightarrow{" + B + "}", "\\overleftrightarrow{" + C + "}" ]) randFromArray([ "\\overline{" + A + B + "}", "\\overline{" + B + A + "}", "\\overline{" + C + B + "}", "\\overline{" + B + C + "}", "\\overline{" + C + A + "}", "\\overline{" + A + C + "}" ]) randFromArray([ "\\overleftrightarrow{" + E + C + "}", "\\overleftrightarrow{" + C + E + "}", "\\overleftrightarrow{" + D + A + "}", "\\overleftrightarrow{" + D + B + "}", "\\overleftrightarrow{" + D + C + "}", "\\overleftrightarrow{" + A + D + "}" ])

What is another way to name line \ell?

SOLUTION

INCORRECT_1
INCORRECT_2
INCORRECT_3
\overleftrightarrow{A + B + C}

Lines are named using any two points on the line. The order doesn't matter.

The points must have the \leftrightarrow above because we're naming a line, not a ray or a segment.

Another way to name line \ell is SOLUTION.

$._("Yes") $._("No") randFromArray([ [[A, B, C], true], [[A, B, D], false], [[A, C, B], true], [[A, C, E], false], [[B, A, C], true], [[B, C, A], true], [[B, E, C], false], [[C, A, B], true], [[C, A, D], false], [[C, B, A], true], [[C, B, E], false], [[E, C, A], false] ])

Are the points toSentence(POINTS) collinear?

SOLUTION ? YES : NO

YES
NO

Collinear means that they lie on the same line.

Can you draw a straight line through points toSentence(POINTS)?

Yes, points toSentence(POINTS) are collinear.

No, points toSentence(POINTS) are not collinear.

$._("Yes") $._("No") shuffle([A, B, C, D, E], 2)

Are the points toSentence(POINTS) collinear?

YES

YES
NO

Through any two points, there is exactly one line.

Points can be collinear even if the line isn't drawn in the figure.

Can you draw a straight line through points toSentence(POINTS)? Actually, can you draw a straight line through any two points?

Yes, points toSentence(POINTS) are collinear.

$._("Yes") $._("No") randFromArray([ [[A, B, C, E], true, []], [[A, B, D, E], false, [A, B, E]], [[A, C, D], true, []], [[A, C, D, E], false, [A, C, E]], [[B, A, C], true, []], [[B, C, A, E], true, []], [[E, C, D, A], false, [E, C, A]], [[B, E, D, C], false, [B, E, C]], [[C, B, D, E], false, [C, B, E]], [[C, D, A], true, []], [[E, D, A], true, []] ])

Are the points toSentence(POINTS) coplanar?

SOLUTION ? YES : NO

YES
NO

Coplanar points are points that all lie on the same plane.

Can a flat surface pass through all the points without bending?

No, any flat surface that includes three of the points won't include the fourth. For example, points toSentence(COPLANAR) are in plane \mathcal{R}, but point D is not.

Yes, points toSentence(POINTS) all lie within a single flat surface. In this case, plane \mathcal{R}.

Yes, there is always at least one flat surface that passes through any three points.