Question

rectangle r with side lengths 4 and 6 is dilated by a scale factor of 2. what are the side lengths of the resulting rectangle?
a. 8 and 12
b. 2 and 3
c. 6 and 9
d. 10 and 15

a triangle with side lengths 5, 12, and 13 is dilated to create a similar triangle with side lengths 30, 72, and 78. what is the dilation factor?
a. 6
b. 10
c. 1/2
d. 1/6

triangle △abc has side lengths 3, 4, and 5. triangle △def has side lengths 6, 8, and 10. are these triangles similar? if so, by what transformation?
a. yes, by dilation.
b. yes, by a rotation.
c. no, the figures are not congruent.
d. yes, by a translation.

a rectangle has vertices a(1, 2), b(5, 2), c(5, 4), and d(1, 4). it is translated to a(0, 0), b(4, 0), c(4, 2), and d(0, 2). what is the translation rule?
a. (x - 1, y + 2)
b. (x + 1, y + 2)
c. (x - 2, y - 2)
d. (x - 1, y - 2)

Explanation:

Step1: Dilate rectangle side lengths

When a figure is dilated by a scale factor \(k\), each side length is multiplied by \(k\). For the rectangle with side lengths \(a = 4\) and \(b=6\) and \(k = 2\), the new side lengths are \(a'=4\times2 = 8\) and \(b'=6\times2=12\).

Step2: Find dilation factor for triangle

The dilation factor \(k\) is the ratio of corresponding side lengths of similar figures. If the original side lengths are \(s_1 = 5\), \(s_2=12\), \(s_3 = 13\) and the new side lengths are \(S_1=30\), \(S_2 = 72\), \(S_3=78\). Using \(k=\frac{S_1}{s_1}=\frac{30}{5}=6\) (checking \(\frac{72}{12}=6\) and \(\frac{78}{13}=6\) for consistency).

Step3: Check triangle similarity

For \(\triangle ABC\) with side lengths \(3\), \(4\), \(5\) and \(\triangle DEF\) with side lengths \(6\), \(8\), \(10\). The ratios are \(\frac{6}{3}=2\), \(\frac{8}{4}=2\), \(\frac{10}{5}=2\). Since all ratios are equal, they are similar by dilation (rotation and translation do not change side - length ratios in the way shown here; congruent is a special case of similarity with ratio \(1\), but these are not congruent as side lengths differ).

Step4: Determine translation rule

For a point \((x,y)\) translated to \((x',y')\). Take point \(A(1,2)\) translated to \(A'(0,0)\). The \(x\) - coordinate change: \(x'=x - 1\) (since \(0=1-1\)), the \(y\) - coordinate change: \(y'=y - 2\) (since \(0 = 2-2\)). Check with \(B(5,2)\): \(x=5\), \(x'=5 - 1=4\); \(y = 2\), \(y'=2-2=0\) (matches \(B'(4,0)\)).

Answer:

a. 8 and 12; a. 6; a. Yes, by dilation; d. \((x - 1,y - 2)\)