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.