Question

graph the image of square qrst after a dilation with a scale factor of $\frac{1}{3}$, centered at the origin.

Explanation:

Step1: Find coordinates of original square

Assume \( Q(-9,-9)\), \( R(9,-9)\), \( S(9,9)\), \( T(-9,9)\)

Step2: Apply dilation formula

For a point \((x,y)\) dilated with scale factor \(k = \frac{1}{3}\) centered at origin \((0,0)\), new coordinates \((x',y')=(k\times x,k\times y)\)

• For \(Q(-9,-9)\): \(x'=\frac{1}{3}\times(-9)= - 3\), \(y'=\frac{1}{3}\times(-9)=-3\), new point \(Q'(-3,-3)\)
• For \(R(9,-9)\): \(x'=\frac{1}{3}\times9 = 3\), \(y'=\frac{1}{3}\times(-9)=-3\), new point \(R'(3,-3)\)
• For \(S(9,9)\): \(x'=\frac{1}{3}\times9 = 3\), \(y'=\frac{1}{3}\times9 = 3\), new point \(S'(3,3)\)
• For \(T(-9,9)\): \(x'=\frac{1}{3}\times(-9)=-3\), \(y'=\frac{1}{3}\times9 = 3\), new point \(T'(-3,3)\)

Step3: Plot new points

Plot \(Q'(-3,-3)\), \(R'(3,-3)\), \(S'(3,3)\), \(T'(-3,3)\) and connect them to form the dilated square.

Answer:

The new square has vertices \(Q'(-3,-3)\), \(R'(3,-3)\), \(S'(3,3)\), \(T'(-3,3)\). Plot these points on the coordinate - plane and connect them.