Depth · one concept from first principles · step 2 of 4
No triangle: a parallel pair or one common point
Three different lines are x-y=0, 2x+y=3, and mx-4y=16. For which values of m do they fail to enclose a triangle?
ExpressionsFunctions
Your answer
How to solve
Strategy Make a Systematic List — List every way a triangle can fail: some pair of lines is parallel (that pair never meets), or all three pass through one point (the three corners collapse). Line 3 is the only one that moves with m, so check it against each fixed line and against their crossing point.
1STEP 1
Write each line in slope form
Solve each equation for y to read off the slopes.
y=x, y=-2x+3, y=(m/4)x-4
Slope form makes parallel lines easy to spot.
8.F.A.3Convert To Algebra2STEP 2
Case 1: line 3 parallel to line 1
Parallel lines have equal slopes and different intercepts (-4 ≠ 0).
m/4=1, m=4
Two parallel lines never meet, so one corner is missing.
8.EE.C.8Make A Systematic List3STEP 3
Case 2: line 3 parallel to line 2
Again equal slopes, and the intercepts -4 and 3 differ.
m/4=-2, m=-8
Lines 1 and 2 have different slopes, so this list of parallel pairs is complete.
8.EE.C.8Make A Systematic List4STEP 4
Case 3: all three through one point
Lines 1 and 2 cross where x=-2x+3, so x=1 and y=1. Line 3 must pass through (1, 1).
m × 1-4 × 1=16, m=20
If the third line runs through the crossing of the other two, all three corners are the same point.
8.EE.C.8Draw A DiagramWhen line 3 passes through , the three lines cannot form a triangle.
Why?
If line 3 also passes through , every pair of lines meets at that one point instead of at three different corners.
Why?
makes , and line 3's equation all true, so all three lines pass through it.
🧱A solution makes the equation trueA solution is a number that, put in for the unknown, makes both sides come out equal; that is all being a solution means.
Why?
Two different lines cross at most once, so is the only crossing of each pair and the three corners collapse into it.
🧱Two different lines cross at most onceTwo points fix a straight line, so two lines that shared two points would be the same line; different lines cross once or, if parallel, never.
Answer
m=-8, 4, 20
Check each value: m=4 gives y=x-4, parallel to y=x; m=-8 gives y=-2x-4, parallel to y=-2x+3; m=20 gives y=5x-4, and 5 × 1-4=1, so it passes through (1, 1). In every case the three lines are still distinct.
Takeaway
Three lines miss a triangle only two ways: two run parallel, or all three meet at one point.
- Write each line in slope form
- Case 1: line 3 parallel to line 1
- Case 2: line 3 parallel to line 2
- Case 3: all three through one point
Where next?
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