Question

the smaller triangle is a pre-image of the bigger triangle. is (1,-1). what is the scale factor used to create the dilation? graph of a coordinate plane with two triangles (smaller pre - image, bigger image) and options: -4, 4, \\(\frac{1}{4}\\), \\(-\frac{1}{4}\\)

Explanation:

Step1: Identify corresponding points

Let's take a vertex of the smaller triangle and its image in the bigger triangle. For example, the vertex of the smaller triangle at \((1,1)\) (wait, maybe better to check the horizontal side). The smaller triangle's horizontal side length: from \(x = -1\) to \(x = 1\), so length \(1 - (-1)=2\)? Wait, no, looking at the grid, the smaller triangle's base (horizontal) is from \((-1, -1)\) to \((1, -1)\)? Wait, the center is \((1,-1)\)? Wait, the problem says "the center is \((1,-1)\)". So dilation center is \((1,-1)\). Let's take a point on the smaller triangle: say \((1,1)\) (top vertex) and its image in the bigger triangle. Wait, maybe the vertical side: smaller triangle's vertical side from \((1,1)\) to \((1,-1)\), length is \(1 - (-1)=2\). The bigger triangle's vertical side: from \((1,1)\) to \((1,-9)\)? Wait, no, the bottom vertex of the bigger triangle is \((1,-9)\)? Wait, the grid: the smaller triangle has a vertex at \((1,1)\), \((-1,-1)\), and \((1,-1)\). The bigger triangle has vertices at \((1,1)\), \((9,-1)\), and \((1,-9)\). Wait, distance from center \((1,-1)\) to \((1,1)\) is \(1 - (-1)=2\) (vertical distance). Distance from center \((1,-1)\) to \((1,-9)\) is \(-1 - (-9)=8\) (vertical distance). Wait, scale factor is image distance / pre - image distance. So \(8/2 = 4\). Alternatively, horizontal: from center \((1,-1)\) to \((-1,-1)\) is \(1 - (-1)=2\) (horizontal distance, pre - image). From center \((1,-1)\) to \((9,-1)\) is \(9 - 1 = 8\) (horizontal distance, image). So scale factor \(= 8/2=4\). Also, check the sign: since the image is larger and in the same general direction (dilation center is \((1,-1)\), the bigger triangle is a dilation of the smaller one, and the lengths are positive multiples, so scale factor is positive 4.

Step2: Calculate scale factor

Scale factor \(k=\frac{\text{length of image segment}}{\text{length of pre - image segment}}\). Taking the horizontal segment from the center \((1,-1)\): pre - image length (from \((1,-1)\) to \((-1,-1)\)) is \(|1 - (-1)| = 2\). Image length (from \((1,-1)\) to \((9,-1)\)) is \(|9 - 1| = 8\). Then \(k=\frac{8}{2}=4\).

Answer:

4