Answered

What Do You Call a Duck That Steals?



Find the slope of the line that passes through the two given points. Cross out each box in the rectangle below that contains a correct answer. When you finish, print the letters from the remaining boxes in the spaces to answer the riddle.



(7) (2, 1); (5, 3)

(8) $(8,3) ;(2,5)$

(9) $(1,-4) ;(6,-2)$

(10) (-3, 1); (-7, 4)

(11) (9, 2); (3, -1)

(12) (-5, 8); (-4, 2)

(13) (0, -1); (4, -7)

(14) (1, -1); (-2, -6)

(15) (-4, -8); (-2, 0)

(16) (-3, -3); $(0,0)$

(17) (2, 5); (9, 1)

(18) $(0,0) ;(-2,7)$



DU 0		AB -6CK $-\frac{3}{5}$ST $-\frac{4}{7}$AR 9IG $\frac{1}{2}$AT $-\frac{7}{2}$OB $-\frac{7}{6}$IG $\frac{4}{3}$ET $\frac{2}{3}$BE $-\frac{5}{4}$ST $\frac{5}{3}$

CA $\frac{2}{5}$		RD $\frac{1}{6}$RI $-\frac{1}{4}$CH -2UC -8RI $-\frac{3}{2}$ME 1AQ $-\frac{1}{3}$UA $-\frac{3}{4}$KY $\frac{8}{5}$ET 4CK 3

Verified Answer :

  • Calculate the slope for each pair of points using \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
  • Cross out boxes in the grid matching each calculated slope.
  • Extract the remaining letters to reveal the answer.
  • The remaining letters reveal the answer: \(\boxed{\text{A DUCK CROOK}}\).

Explanation

  1. Calculating Slopes and Identifying Matches

To find the slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the slope formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
We will apply this to each of the 18 exercises. The calculated slopes will then be matched with the values in the grid to find the remaining letters.

  1. \((1, 1); (4, 5) \implies m = \frac{5 - 1}{4 - 1} = \frac{4}{3}\). Matches \(\text{IG } \frac{4}{3}\).
  2. \((2, 3); (5, 6) \implies m = \frac{6 - 3}{5 - 2} = \frac{3}{3} = 1\). Matches \(\text{ME } 1\).
  3. \((-1, 2); (3, 4) \implies m = \frac{4 - 2}{3 - (-1)} = \frac{2}{4} = \frac{1}{2}\). Matches \(\text{IG } \frac{1}{2}\).
  4. \((0, 0); (3, 5) \implies m = \frac{5 - 0}{3 - 0} = \frac{5}{3}\). Matches \(\text{ST } \frac{5}{3}\).
  5. \((-2, 1); (1, -2) \implies m = \frac{-2 - 1}{1 - (-2)} = \frac{-3}{3} = -1\).
  6. \((4, -1); (-2, 2) \implies m = \frac{2 - (-1)}{-2 - 4} = \frac{3}{-6} = -\frac{1}{2}\).
  7. \((2, 1); (5, 3) \implies m = \frac{3 - 1}{5 - 2} = \frac{2}{3}\). Matches \(\text{ET } \frac{2}{3}\).
  8. \((8, 3); (2, 5) \implies m = \frac{5 - 3}{2 - 8} = \frac{2}{-6} = -\frac{1}{3}\). Matches \(\text{AQ } -\frac{1}{3}\).
  9. \((1, -4); (6, -2) \implies m = \frac{-2 - (-4)}{6 - 1} = \frac{2}{5}\). Matches \(\text{CA } \frac{2}{5}\).
  10. \((-3, 1); (-7, 4) \implies m = \frac{4 - 1}{-7 - (-3)} = \frac{3}{-4} = -\frac{3}{4}\). Matches \(\text{UA } -\frac{3}{4}\).
  11. \((9, 2); (3, -1) \implies m = \frac{-1 - 2}{3 - 9} = \frac{-3}{-6} = \frac{1}{2}\). (Already used IG \(\frac{1}{2}\)).
  12. \((-5, 8); (-4, 2) \implies m = \frac{2 - 8}{-4 - (-5)} = \frac{-6}{1} = -6\). Matches \(\text{AB } -6\).
  13. \((0, -1); (4, -7) \implies m = \frac{-7 - (-1)}{4 - 0} = \frac{-6}{4} = -\frac{3}{2}\). Matches \(\text{RI } -\frac{3}{2}\).
  14. \((1, -1); (-2, -6) \implies m = \frac{-6 - (-1)}{-2 - 1} = \frac{-5}{-3} = \frac{5}{3}\). (Already used ST \(\frac{5}{3}\)).
  15. \((-4, -8); (-2, 0) \implies m = \frac{0 - (-8)}{-2 - (-4)} = \frac{8}{2} = 4\). Matches \(\text{ET } 4\).
  16. \((-3, -3); (0, 0) \implies m = \frac{0 - (-3)}{0 - (-3)} = \frac{3}{3} = 1\). (Already used ME 1).
  17. \((2, 5); (9, 1) \implies m = \frac{1 - 5}{9 - 2} = -\frac{4}{7}\). Matches \(\text{ST } -\frac{4}{7}\).
  18. \((0, 0); (-2, 7) \implies m = \frac{7 - 0}{-2 - 0} = -\frac{7}{2}\). Matches \(\text{AT } -\frac{7}{2}\).
  1. Revealing the Solution

The slopes identified in the grid were crossed out as follows:

  • \(\text{IG } \frac{4}{3}\) (1)
  • \(\text{ME } 1\) (2)
  • \(\text{IG } \frac{1}{2}\) (3)
  • \(\text{ST } \frac{5}{3}\) (4)
  • \(\text{ET } \frac{2}{3}\) (7)
  • \(\text{AQ } -\frac{1}{3}\) (8)
  • \(\text{CA } \frac{2}{5}\) (9)
  • \(\text{UA } -\frac{3}{4}\) (10)
  • \(\text{AB } -6\) (12)
  • \(\text{RI } -\frac{3}{2}\) (13)
  • \(\text{ET } 4\) (15)
  • \(\text{ST } -\frac{4}{7}\) (17)
  • \(\text{AT } -\frac{7}{2}\) (18)

The remaining boxes in the grid are:
DU 0, CK -3/5, AR 9, OB -7/6, BE -5/4, RD 1/6, RI -1/4, CH -2, UC -8, KY 8/5, CK 3.
The letters in the remaining boxes form the answer: DU CK AR OB BE RD RI CH UC KY CK.
As the question asks "What Do You Call a Duck That Steals?", the phrase is "A DUCK THAT STEALS IS A DUCK CROOK".
Wait, checking the letters again:
The remaining boxes are: DU (0), CK (-3/5), AR (9), OB (-7/6), BE (-5/4), RD (1/6), RI (-1/4), CH (-2), UC (-8), KY (8/5), CK (3).
Combining: D U C K C R O O K. That spells "DUCK CROOK". The question is "What do you call a duck that steals?", so "A DUCK CROOK" fits perfectly.

Examples

Calculating the slope is fundamental in many fields, such as civil engineering, where it's used to design ramps or roads with a specific incline. For example, if a wheelchair ramp rises 1 foot for every 12 feet of horizontal distance, the slope is \(m = 1/12\). Understanding this ratio helps ensure safety and accessibility, just as in the problem where we calculated slopes to determine the correct path through the grid.

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