Question

Quadrilateral JKLM is mapped to quadrilateral J'K'L'M' using the dilation (x, y)→(3x/2, 3y/2). Then quadrilateral J'K'L'M' is mapped to quadrilateral J''K''L''M'' using the translation (x, y)→(x + 3, y - 4). The vertices of quadrilateral J'K'L'M' are J(-12, 0), K(-12, 18), L(-6, 18), and M(-6, 0). Find the coordinates of the vertices of quadrilateral JKLM and quadrilateral J''K''L''M''. Are quadrilateral JKLM and quadrilateral J''K''L''M'' similar? Explain.

Answer 1

The coordinates of quadrilateral JKLM are obtained by reversing the dilation applied to J'K'L'M'. Quadrilateral J''K''L''M'' coordinates are found using translation. Both quadrilaterals are similar as dilation and translation preserve shape.

Step-by-step explanation:

To find the coordinates of the original quadrilateral JKLM, we need to reverse the dilation that was applied to get quadrilateral J'K'L'M'. The scale factor for the dilation was ⅓ (3/2 for both x and y coordinates), so we will multiply the coordinates of J'K'L'M' by ¾ (the reciprocal of ⅓) to get the coordinates of JKLM.

The coordinates for quadrilateral JKLM are then found as:
J (-12 * ¾, 0 * ¾) = J (-9, 0)
K (-12 * ¾, 18 * ¾) = K (-9, 13.5)
L (-6 * ¾, 18 * ¾) = L (-4.5, 13.5)
M (-6 * ¾, 0 * ¾) = M (-4.5, 0)

To find the coordinates of quadrilateral J''K''L''M'', apply the given translation (x + 3, y - 4) to each of the points of J'K'L'M':
J' (-12 + 3, 0 - 4) = J'' (-9, -4)
K' (-12 + 3, 18 - 4) = K'' (-9, 14)
L' (-6 + 3, 18 - 4) = L'' (-3, 14)
M' (-6 + 3, 0 - 4) = M'' (-3, -4)

Since both the dilation transformation and the translation transformation preserve the shape of geometrical figures, quadrilaterals JKLM and J''K''L''M'' are similar. This is because dilation is a similarity transformation that changes the size without changing the shape, and translation moves the figure without changing its size or shape.