Quadrilateral $$ J K L M $$ has vertices at $$ J(0,0), K(0,6) $$, $$ L(9,12) $$, and $$ M(12,0) $$. Find the vertices of the quadrilateral after a dilation with scale factor $$ \frac{2}{3} $$. $$ \begin{array}{l}J^{\prime}(0,0), K^{\prime}(0,9), L^{\prime}(27 / 2,18) ext {, and } M^{\prime}(18,0) \ J^{\prime}(0,0), K^{\prime}(0,4), L^{\prime}(6,8) ext {, and } M^{\prime}(8,0) \ J^{\prime}(0,0), K^{\prime}(0,-4), L^{\prime}(-6,-4) ext {, and } M^{\prime}(-8,0) \ J^{\prime}(2 / 3,2 / 3), K^{\prime}(2 / 3,20 / 3), L^{\prime}(9 / 3,38 / 3) ext {, and } \ M^{\prime}(38 / 3,2 / 3)\end{array} $$

Question

Quadrilateral $$ J K L M $$ has vertices at $$ J(0,0), K(0,6) $$, $$ L(9,12) $$, and $$ M(12,0) $$. Find the vertices of the quadrilateral after a dilation with scale factor $$ \frac{2}{3} $$. $$ \begin{array}{l}J^{\prime}(0,0), K^{\prime}(0,9), L^{\prime}(27 / 2,18) 	ext {, and } M^{\prime}(18,0) \ J^{\prime}(0,0), K^{\prime}(0,4), L^{\prime}(6,8) 	ext {, and } M^{\prime}(8,0) \ J^{\prime}(0,0), K^{\prime}(0,-4), L^{\prime}(-6,-4) 	ext {, and } M^{\prime}(-8,0) \ J^{\prime}(2 / 3,2 / 3), K^{\prime}(2 / 3,20 / 3), L^{\prime}(9 / 3,38 / 3) 	ext {, and } \ M^{\prime}(38 / 3,2 / 3)\end{array} $$

UpStudy AI · Accepted Answer

To find the vertices of quadrilateral $$ J K L M $$ after a dilation with a scale factor of $$ \frac{2}{3} $$ centered at the origin $$(0,0)$$, we apply the dilation to each vertex individually. The dilation formula for a point $$(x, y)$$ with scale factor $$ k $$ centered at the origin is:

$$
(x', y') = \left( kx, ky \right)
$$

Given $$ k = \frac{2}{3} $$, let's apply this to each vertex:

1. **Vertex $$ J(0,0) $$:**
   $$
   J' = \left( \frac{2}{3} \times 0, \frac{2}{3} \times 0 \right) = (0, 0)
   $$

2. **Vertex $$ K(0,6) $$:**
   $$
   K' = \left( \frac{2}{3} \times 0, \frac{2}{3} \times 6 \right) = \left( 0, 4 \right)
   $$

3. **Vertex $$ L(9,12) $$:**
   $$
   L' = \left( \frac{2}{3} \times 9, \frac{2}{3} \times 12 \right) = (6, 8)
   $$

4. **Vertex $$ M(12,0) $$:**
   $$
   M' = \left( \frac{2}{3} \times 12, \frac{2}{3} \times 0 \right) = (8, 0)
   $$

After applying the dilation, the new vertices of the quadrilateral are:
$$
J'(0,0), \quad K'(0,4), \quad L'(6,8), \quad \text{and} \quad M'(8,0)
$$

**Answer:** 
$$ J^{\prime}(0,0), K^{\prime}(0,4), L^{\prime}(6,8) \text {, and } M^{\prime}(8,0) $$
After a dilation with a scale factor of $$ \frac{2}{3} $$, the vertices of quadrilateral $$ J K L M $$ are: - $$ J'(0,0) $$ - $$ K'(0,4) $$ - $$ L'(6,8) $$ - $$ M'(8,0) $$

Quadrilateral has vertices at ,
, and . Find the vertices of the quadrilateral
after a dilation with scale factor .

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Quadrilateral has vertices at , , and . Find the vertices of the quadrilateral after a dilation with scale factor .