Question

2.5 puzzle time
what do you call a surgeon with eight arms?
write the letter of each answer in the box containing the exercise number.
the vertices of a triangle are a(2, 2), b(2, 5), and c(4, 2). find the coordinates of the image after a dilation with the given scale factor.

1. ( k = 2 ) 2. ( k = 5 ) 3. ( k = 1\frac{1}{2} )

the vertices of a triangle are a(-5, 5), b(-2, -5), and c(-2, 0) find the coordinates of the image after a dilation with the given scale factor.

1. ( k = \frac{1}{2} ) 5. ( k = 0.75 ) 6. ( k = \frac{1}{5} )

the vertices of a triangle are a(1, 3), b(7, 3), and c(7, 5). the vertices of its image after a dilation are given. find the scale factor.

1. ( a(4, 12), b(28, 12), c(28, 20) )
2. ( aleft( \frac{1}{6}, \frac{1}{2}

ight), bleft( 1\frac{1}{6}, \frac{1}{2}
ight), cleft( 1\frac{1}{6}, \frac{5}{6}
ight) )

1. ( a(0.5, 1.5), b(3.5, 1.5), c(3.5, 2.5) )

answers
b. ( k = \frac{1}{4} )
d. ( a(3, 3), bleft( 3, 7\frac{1}{2}
ight), c(6, 3) )
t. ( a(-3.75, 3.75), b(-1.5, -3.75), c(-1.5, 0) )
e. ( k = 2 )
c. ( a(4, 4), b(4, 10), c(8, 4) )
o. ( k = 4 )
a. ( k = 0.5 )
s. ( aleft( -2\frac{1}{2}, -2\frac{1}{2}
ight), b(-1, -2\frac{1}{2}), c(-1, 0) )
f. ( k = \frac{1}{7} )
p. ( k = \frac{1}{6} )
u. ( a(10, 10), b(10, 25), c(20, 10) )
j. ( k = 6 )
o. ( a(-1, 1), bleft( -\frac{2}{5}, -1
ight), cleft( -\frac{2}{5}, 0
ight) )
t. ( k = 7 )

Explanation:

Step1: Dilation formula

For a point \((x,y)\) and scale factor \(k\), the image after dilation is \((kx,ky)\).

Step2: Solve for \(k = 2\)

For \(A(2,2)\), \(A'=(2\times2,2\times2)=(4,4)\); for \(B(2,5)\), \(B'=(2\times2,5\times2)=(4,10)\); for \(C(4,2)\), \(C'=(4\times2,2\times2)=(8,4)\). This matches option \(C\).

Step3: Solve for \(k = 5\)

For \(A(2,2)\), \(A'=(2\times5,2\times5)=(10,10)\); for \(B(2,5)\), \(B'=(2\times5,5\times5)=(10,25)\); for \(C(4,2)\), \(C'=(4\times5,2\times5)=(20,10)\). This matches option \(U\).

Step4: Solve for \(k = 1\frac{1}{2}=\frac{3}{2}\)

For \(A(2,2)\), \(A'=(2\times\frac{3}{2},2\times\frac{3}{2})=(3,3)\); for \(B(2,5)\), \(B'=(2\times\frac{3}{2},5\times\frac{3}{2})=(3,7\frac{1}{2})\); for \(C(4,2)\), \(C'=(4\times\frac{3}{2},2\times\frac{3}{2})=(6,3)\). This matches option \(D\).

Step5: Solve for \(k=\frac{1}{2}\)

For \(A(-5,5)\), \(A'=(-5\times\frac{1}{2},5\times\frac{1}{2})=(-2\frac{1}{2},2\frac{1}{2})\); for \(B(-2,-5)\), \(B'=(-2\times\frac{1}{2},-5\times\frac{1}{2})=(-1,-2\frac{1}{2})\); for \(C(-2,0)\), \(C'=(-2\times\frac{1}{2},0\times\frac{1}{2})=(-1,0)\). This matches option \(S\).

Step6: Solve for \(k = 0.75=\frac{3}{4}\)

For \(A(-5,5)\), \(A'=(-5\times\frac{3}{4},5\times\frac{3}{4})=(-3.75,3.75)\); for \(B(-2,-5)\), \(B'=(-2\times\frac{3}{4},-5\times\frac{3}{4})=(-1.5,-3.75)\); for \(C(-2,0)\), \(C'=(-2\times\frac{3}{4},0\times\frac{3}{4})=(-1.5,0)\). This matches option \(T\).

Step7: Solve for \(k=\frac{1}{5}\)

For \(A(-5,5)\), \(A'=(-5\times\frac{1}{5},5\times\frac{1}{5})=(-1,1)\); for \(B(-2,-5)\), \(B'=(-2\times\frac{1}{5},-5\times\frac{1}{5})=(-\frac{2}{5},-1)\); for \(C(-2,0)\), \(C'=(-2\times\frac{1}{5},0\times\frac{1}{5})=(-\frac{2}{5},0)\). This matches option \(O\).

Step8: Find scale factor for \(A'(4,12)\), \(B'(28,12)\), \(C'(28,20)\)

Take \(A(1,3)\) and \(A'(4,12)\), \(k=\frac{4}{1} = 4\) (check \(y\)-coordinate \(\frac{12}{3}=4\)). This matches option \(O\) (scale - factor \(k = 4\)).

Step9: Find scale factor for \(A'(\frac{1}{6},\frac{1}{2})\), \(B'(1\frac{1}{6},\frac{1}{2})\), \(C'(1\frac{1}{6},\frac{5}{6})\)

Take \(A(1,3)\) and \(A'(\frac{1}{6},\frac{1}{2})\), \(k=\frac{\frac{1}{6}}{1}=\frac{1}{6}\) (check \(y\)-coordinate \(\frac{\frac{1}{2}}{3}=\frac{1}{6}\)). This matches option \(P\).

Step10: Find scale factor for \(A'(0.5,1.5)\), \(B'(3.5,1.5)\), \(C'(3.5,2.5)\)

Take \(A(1,3)\) and \(A'(0.5,1.5)\), \(k=\frac{0.5}{1}=0.5\) (check \(y\)-coordinate \(\frac{1.5}{3}=0.5\)). This matches option \(A\).

Answer:

1. C, 2. U, 3. D, 4. S, 5. T, 6. O, 7. O, 8. P, 9. A