Question

find the coordinates of the vertices of each figure after the given transformation.

1. dilation of $\frac{3}{2}$ about the origin
2. dilation of 1.5 about the origin
3. dilation of $\frac{5}{2}$ about the origin
4. dilation of 5 about the origin
5. dilation of 1.5 about the origin
6. dilation of 2 about the origin

Answer:

1. Explanation for dilation about the origin:

• When a point \((x,y)\) is dilated about the origin by a scale - factor \(k\), the new coordinates \((x',y')\) are given by the rule \((x',y')=(k x,k y)\).

1. For problem 13 (assuming we can identify the vertices of \(\triangle STU\)):

• Let the vertices of \(\triangle STU\) be \(S(x_1,y_1)\), \(T(x_2,y_2)\), and \(U(x_3,y_3)\).
• Step 1: Find the new \(x\) - coordinates
• The new \(x\) - coordinates of the vertices are \(x_1'=\frac{3}{2}x_1\), \(x_2'=\frac{3}{2}x_2\), \(x_3'=\frac{3}{2}x_3\).
• Step 2: Find the new \(y\) - coordinates
• The new \(y\) - coordinates of the vertices are \(y_1'=\frac{3}{2}y_1\), \(y_2'=\frac{3}{2}y_2\), \(y_3'=\frac{3}{2}y_3\).
• For example, if \(S=(2, - 1)\), \(T=(2,2)\), \(U=(4,2)\):
• \(S'=(\frac{3}{2}\times2,\frac{3}{2}\times(-1))=(3,-\frac{3}{2})\)
• \(T'=(\frac{3}{2}\times2,\frac{3}{2}\times2)=(3,3)\)
• \(U'=(\frac{3}{2}\times4,\frac{3}{2}\times2)=(6,3)\)

1. For problem 14 (assuming we can identify the vertices of \(\triangle IJO\)):

• Let the vertices of \(\triangle IJO\) be \(I(x_4,y_4)\), \(J(x_5,y_5)\), \(O(x_6,y_6)\).
• Step 1: Apply the dilation formula for \(x\) - coordinates
• \(x_4' = 1.5x_4\), \(x_5' = 1.5x_5\), \(x_6' = 1.5x_6\).
• Step 2: Apply the dilation formula for \(y\) - coordinates
• \(y_4' = 1.5y_4\), \(y_5' = 1.5y_5\), \(y_6' = 1.5y_6\).

1. For problem 15 (assuming we can identify the vertices of \(\triangle DEF\)):

• Let the vertices of \(\triangle DEF\) be \(D(x_7,y_7)\), \(E(x_8,y_8)\), \(F(x_9,y_9)\).
• Step 1: Calculate new \(x\) - coordinates
• \(x_7'=\frac{5}{2}x_7\), \(x_8'=\frac{5}{2}x_8\), \(x_9'=\frac{5}{2}x_9\).
• Step 2: Calculate new \(y\) - coordinates
• \(y_7'=\frac{5}{2}y_7\), \(y_8'=\frac{5}{2}y_8\), \(y_9'=\frac{5}{2}y_9\).

1. For problem 16 (assuming we can identify the vertices of \(\triangle MKL\)):

• Let the vertices of \(\triangle MKL\) be \(M(x_{10},y_{10})\), \(K(x_{11},y_{11})\), \(L(x_{12},y_{12})\).
• Step 1: Find new \(x\) - coordinates
• \(x_{10}' = 5x_{10}\), \(x_{11}' = 5x_{11}\), \(x_{12}' = 5x_{12}\).
• Step 2: Find new \(y\) - coordinates
• \(y_{10}' = 5y_{10}\), \(y_{11}' = 5y_{11}\), \(y_{12}' = 5y_{12}\).

1. For problem 17 (assuming we can identify the vertices of \(\triangle XYW\)):

• Let the vertices of \(\triangle XYW\) be \(X(x_{13},y_{13})\), \(Y(x_{14},y_{14})\), \(W(x_{15},y_{15})\).
• Step 1: Determine new \(x\) - coordinates
• \(x_{13}' = 1.5x_{13}\), \(x_{14}' = 1.5x_{14}\), \(x_{15}' = 1.5x_{15}\).
• Step 2: Determine new \(y\) - coordinates
• \(y_{13}' = 1.5y_{13}\), \(y_{14}' = 1.5y_{14}\), \(y_{15}' = 1.5y_{15}\).

1. For problem 18 (assuming we can identify the vertices of \(\triangle FGH\)):

• Let the vertices of \(\triangle FGH\) be \(F(x_{16},y_{16})\), \(G(x_{17},y_{17})\), \(H(x_{18},y_{18})\).
• Step 1: Compute new \(x\) - coordinates
• \(x_{16}' = 2x_{16}\), \(x_{17}' = 2x_{17}\), \(x_{18}' = 2x_{18}\).
• Step 2: Compute new \(y\) - coordinates
• \(y_{16}' = 2y_{16}\), \(y_{17}' = 2y_{17}\), \(y_{18}' = 2y_{18}\).

Since we don't have the actual coordinates of the vertices from the image, we can't give the exact final - answer coordinates. But the general method for finding the coordinates of the vertices of a figure after dilation about the origin is as described above. If you provide the coordinates of the vertices of each figure, we can calculate the exact new coordinates.