Depth · one concept from first principles · step 2 of 4
The two lines and meet at the point . A third line is drawn. Find all values of the constant for which the three lines do not enclose a triangle. ( is a constant.)
The line y=mx-1 passes through (4, 7).
Now all three lines are known except a.
8.F.A.3Convert To AlgebraIf y=ax+5 also passes through (4, 7), the three crossing points collapse into one.
Three lines through one point make a star, not a triangle.
8.EE.C.8Convert To AlgebraEqual slopes with different y-intercepts (5 ≠ 3) means parallel, so those two never meet.
A missing corner means no triangle.
8.EE.C.8Make A Systematic ListLikewise, matching the slope m=2 makes the third line parallel to the second (5 ≠ -1).
Again one pair never crosses.
8.EE.C.8Make A Systematic ListAny other a gives two new crossings, different from (4, 7) and from each other, so a triangle forms.
Exactly three bad values.
8.EE.C.8Make A Systematic ListThe three lines fail to enclose a triangle exactly when , or .
A triangle needs three different corners, one where each pair of lines crosses, and two things can spoil that.
If the third line has the slope of one of the others but a different intercept, the two are parallel and never meet, so a corner is missing.
If makes true, all three corners become that one point.
making true means the third line passes through the point where the other two cross.
Two different lines cross at most once, so that shared point is the only crossing of every pair and the three corners collapse into it.
To make a triangle you need three different corners -- so hunt for the ways to lose a corner: parallel lines, or all three crossing at one spot.
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