randFromArray([ [ "A", "B", "C" ], [ "J", "K", "L" ], [ "C", "J", "T" ] ]) POINTS[ 0 ] + POINTS[ 1 ] POINTS[ 1 ] + POINTS[ 2 ] POINTS[ 0 ] + POINTS[ 2 ]
randRange( 1, 9 ) randRangeNonZero( 2, 9 ) randRangeNonZero( -9, 9 ) randRangeNonZero( 2, 9 ) COEF_1 * X + CONST_1 - COEF_2 * X ( COEF_1 * X + CONST_1 ) + ( COEF_2 * X + CONST_2 )

`POINTS[ 1 ]` is the midpoint of `\overline{SEG_TOTAL}`

init({ range: [ [ -1, 11 ], [ -1, 1 ] ] }); line( [ 0, 0 ], [ 10, 0 ] ); style({ stroke: "#000", fill: "#000" }); graph.points = raphael.set(); graph.points.push( circle( [ 0, 0 ], 0.10 ) ); graph.points.push( circle( [ 5, 0 ], 0.10 ) ); graph.points.push( circle( [ 10, 0 ], 0.10 ) ); label( [ 0, 0 ], POINTS[ 0 ], "below" ); label( [ 5, 0 ], POINTS[ 1 ], "below" ); label( [ 10, 0 ], POINTS[ 2 ], "below" );

If:
`\qquad SEG_1 = COEF_1x + CONST_1` and
`\qquad SEG_2 = COEF_2x + CONST_2`

Find `SEG_TOTAL`.

TOTAL

A midpoint divides a segment into two segments with equal lengths.

style({ stroke: BLUE, strokeWidth: 3 }); line( [ 0, 0 ], [ 5, 0 ] ); style({ stroke: GREEN, strokeWidth: 3 }); line( [ 5, 0 ], [ 10, 0 ] ); graph.points.toFront(); $( "#given1" ).css({ "color": BLUE }); $( "#given2" ).css({ "color": GREEN });

`\blue{SEG_1} = \green{SEG_2}`

Substitute in the expressions that were given for each length:

`\qquad \blue{COEF_1x + CONST_1} = \green{COEF_2x + CONST_2}`

Solve for `x`:

`\qquad expr([ "*", COEF_1 - COEF_2, "x" ]) = CONST_2 - CONST_1`

`\qquad x = X`

Substitute `X` for `x` in the expressions that were given for `SEG_1` and `SEG_2`:

`\qquad SEG_1 = COEF_1(\pink{X}) + CONST_1` `\qquad SEG_2 = COEF_2(\pink{X}) + CONST_2`

`\qquad SEG_1 = COEF_1 * X + CONST_1` `\qquad SEG_2 = COEF_2 * X + CONST_2`

`\qquad SEG_1 = COEF_1 * X + CONST_1` `\qquad SEG_2 = COEF_2 * X + CONST_2`

To find the length `SEG_TOTAL`, add the lengths `\blue{SEG_1}` and `\green{SEG_2}`:

`\qquad SEG_TOTAL = \blue{SEG_1} + \green{SEG_2}`

`\qquad SEG_TOTAL = \blue{COEF_1 * X + CONST_1} + \green{COEF_2 * X + CONST_2}`

`\qquad SEG_TOTAL = TOTAL`