Question

dilate triangle hij using (-7, -7) as the center of dilation and a scale factor of 4.

Explanation:

1. First, assume the coordinates of the vertices of triangle \(HIJ\) are \((x_1,y_1)\), \((x_2,y_2)\), \((x_3,y_3)\). Let the center of dilation be \((a,b)=(- 7,-7)\) and the scale - factor \(k = 4\).

• The formula for dilation with center \((a,b)\) and scale - factor \(k\) is \((x,y)\to(a + k(x - a),b + k(y - b))\).

1. Let's assume the coordinates of point \(H\) are \((x_H,y_H)\), for point \(I\) are \((x_I,y_I)\), and for point \(J\) are \((x_J,y_J)\).

• For a general point \((x,y)\) of the triangle, the new coordinates \((x',y')\) after dilation are given by:
• \(x'=-7 + 4(x+7)=-7 + 4x+28=4x + 21\).
• \(y'=-7 + 4(y + 7)=-7+4y + 28=4y + 21\).
• Suppose \(H=(-8,-6)\), \(I=(-7,-5)\), \(J=(-5,-5)\) (by observing the grid).
• For point \(H\):
• \(x_H=-8\), \(y_H=-6\).
• \(x_{H}'=4\times(-8)+21=-32 + 21=-11\).
• \(y_{H}'=4\times(-6)+21=-24 + 21=-3\).
• For point \(I\):
• \(x_I=-7\), \(y_I=-5\).
• \(x_{I}'=4\times(-7)+21=-28 + 21=-7\).
• \(y_{I}'=4\times(-5)+21=-20 + 21=1\).
• For point \(J\):
• \(x_J=-5\), \(y_J=-5\).
• \(x_{J}'=4\times(-5)+21=-20 + 21=1\).
• \(y_{J}'=4\times(-5)+21=-20 + 21=1\).

The new coordinates of the dilated triangle (assuming the above - guessed original coordinates) are \(H'(-11,-3)\), \(I'(-7,1)\), \(J'(1,1)\). But in general, use the formula \((x,y)\to(4x + 21,4y + 21)\) to find the new coordinates of the vertices of the dilated triangle.

Answer:

Use the formula \((x,y)\to(4x + 21,4y + 21)\) to find the new vertices of the dilated triangle, where \((x,y)\) are the original vertices and \((4x + 21,4y + 21)\) are the vertices of the dilated triangle with center of dilation \((-7,-7)\) and scale - factor \(4\).