# Khan/khan-exercises

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 Midpoint formula
randRange( -8, 7 ) randRange( -7, 8 ) randRange( -9, 8 ) randRange( -9, 8 ) ( X1 + X2 ) / 2 ( Y1 + Y2 ) / 2 { color: BLUE, stroke: BLUE, fill: BLUE } { color: GREEN, stroke: GREEN, fill: GREEN, strokeWidth: 1.5 } { color: PURPLE, stroke: PURPLE, fill: PURPLE, strokeWidth: 1.5 }

Point \blue{A} is at \blue{(X1, Y1)} and point \green{B} is at \green{(X2, Y2)}.

What is the midpoint of line segment \overline{AB}?

graphInit({ range: 10, scale: 20, tickStep: 1, labelStep: 1, unityLabels: false, labelFormat: function( s ) { return "\\small{" + s + "}"; }, axisArrows: "<->", }); path([ [ X1, Y1 ], [ X2, Y2 ] ], {stroke:"#000000"}); style(point1Style); label([X1, Y1], "A", getPos([X1, Y1], [X2, Y2])); circle([X1, Y1], 0.15); style(point2Style); label([X2, Y2], "B", getPos([X2, Y2], [X1, Y1])); circle([X2, Y2], 0.15);

(XM, YM)

The x coordinate of the midpoint is the average of the x coordinates of \blue{A} and \green{B}.

x = \dfrac{1}{2}(\blue{X1} + \green{X2})

x = \dfrac{1}{2}(X1 + X2)

x = \purple{XM}

style( midPtStyle ); graph.vert = path( [ [ XM, -10], [ XM, 10] ] );

The y coordinate of the midpoint is the average of the y coordinates \blue{A} and \green{B}.

y = \dfrac{1}{2}(\blue{Y1} + \green{Y2})

y = \dfrac{1}{2}(Y1 + Y2)

y = \purple{YM}

graph.horiz = path( [ [ -10, YM], [ 10, YM ] ] );

The midpoint is (\purple{XM}, \purple{YM}).

circle([XM, YM], 0.15); label([XM, YM], "M", getPos([XM, YM], [X1, Y1], true));
graph.vert.remove(); graph.horiz.remove();

Point \blue{A} is at \blue{(X1, Y1)} and point \purple{M} is at \purple{(XM, YM)}.
Point \purple{M} is the midpoint of point \blue{A} and point \green{B}.

What are the coordinates of point \green{B}?

graphInit({ range: 10, scale: 20, tickStep: 1, labelStep: 1, unityLabels: false, labelFormat: function( s ) { return "\\small{" + s + "}"; }, axisArrows: "<->", }); style(point1Style); label([X1, Y1], "A", getPos([X1, Y1], [XM, YM])); graph.first = circle([X1, Y1], 0.15); style(midPtStyle); label([XM, YM], "M", getPos([XM, YM], [X1, Y1], true)); graph.midpoint = circle([XM, YM], 0.15);

(X2, Y2)

The average of the x coordinates of point \blue{A} and point \green{B} should be \purple{XM}.

\dfrac{1}{2}(\blue{X1} + \green{x}) = \purple{XM}

Solving for \green{x}:

\blue{X1} + \green{x} = 2 * XM

\green{x} = X2

style( point2Style ); graph.vert = path( [ [ X2, -10], [ X2, 10] ] );

The average of the y coordinates of point \blue{A} and point \green{B} should be \purple{YM}.

\dfrac{1}{2}(\blue{Y1} + \green{y}) = \purple{YM}

Solving for \green{y}:

\blue{Y1} + \green{y} = 2 * YM

\green{y} = Y2

graph.horiz = path( [ [ -10, Y2 ], [ 10, Y2 ] ] );

Point \green{B} is (\green{X2}, \green{Y2}).

path([[X1, Y1], [X2, Y2]], {stroke:"#000000"}); circle([X2, Y2], 0.15); label([X2, Y2], "B", getPos([X2, Y2], [XM, YM])); graph.first.toFront(); graph.midpoint.toFront();
graph.vert.remove(); graph.horiz.remove();