When two lines intersect at 90° degrees angle, the lines are perpendicular to each other. Two perpendicular lines, their slope will give a product of -1
i.e. if the first's line slope is 5, then the second line's will be -1 ÷ 5 = -¹/₅
To find the slope of a line, we divide the vertical distance by the horizontal distance.
We'll use the trial and error method to find the right pairing
Let's start with A(3, 3) and B(12, 6)
Vertical distance =
Horizontal distance =
The slope AB = ³/₉ = ¹/₃
We want BC to have a slope -1 ÷ ¹/₃ = -3
Try C(16, -6); check the slope with B(12, 6)
Vertical distance =
Horizontal distance =
Slope of BC = -12 ÷ 4 = -3
The slope BC = -3 is the value we want so, tile 1 pair with tile 4
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Let's do A(-10, 5) and B(12, 16)
Vertical distance = 16 - 5 = 11
Horizontal distance = 12 - -10 = 22
Slope AB = ¹¹/₂₂ = ¹/₂
The perpendicular slope would be -1 ÷ ¹/₂ = -2
Try C(18, 4) with B(12, 16)
Vertical distance = 16 - 4 = 12
Horizontal distance = 12 - 18 = -6
Slope BC = ¹²/₋₆ = -2
Slope BC and slope AB perpendicular, so tile 3 matches with tile 6
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Let's try A(12, -14) and B(-16, 21)
Vertical distance = 21 - -14 = 35
Horizontal distance = -16 - 12 = -28
The slope AB = ³⁵/-₂₈ = ⁵/₋₄
We need the perpendicular slope to be -1 ÷ -⁵/₄ = ⁴/₅
Try C(-11, 25)
Vertical distance with B = 25 - 21 = 4
Horizontal distance with B = -11 - -16 = 5
The slope = ⁴/₅
Tile 7 matches tile 8
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Take A(-12, -19) and B(20, 45)
Vertical distance = 45 - -19 = 64
Horizontal distance = 20 - -12 = 32
Slope AB = ⁶⁴/₃₂ = 2
We need the perpendicular slope to be -1 ÷ 2 = -¹/₂
We have C(6, 52) and checking the slope with B(20, 45)
Vertical distance = 45 - 52 = -7
Horizontal distance = 20 - 6 = 14
The slope is ⁻⁷/₁₄ = -¹/₂
Tile 9 pairs with tile 2
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Conclusion
Tile 1 ⇒ Tile 4
Tile 3 ⇒ Tile 6
Tile 7 ⇒ Tile 8
Tile 9 ⇒ Tile 2
Tile 5 and Tile 10 do not have pairs
= 2.25
= 2.125