Question

1. triangle abc is shown on the coordinate grid below. what is the dilation factor that occurred?

Explanation:

Step1: Select a side - length to compare

Let's consider side BC and B'C'.

Step2: Calculate length of BC

Count the horizontal and vertical distances between B and C. B(-1,2), C(1,1). Horizontal distance $x = 1-(-1)=2$, vertical distance $y=2 - 1 = 1$. Using the distance formula $d=\sqrt{x^{2}+y^{2}}$, $BC=\sqrt{(1 + 1)^{2}+(2 - 1)^{2}}=\sqrt{4 + 1}=\sqrt{5}$.

Step3: Calculate length of B'C'

B'(-2,4), C'(3,3). Horizontal distance $x=3-(-2)=5$, vertical distance $y = 4-3 = 1$. Using the distance formula $d=\sqrt{x^{2}+y^{2}}$, $B'C'=\sqrt{(3 + 2)^{2}+(4 - 3)^{2}}=\sqrt{25+1}=\sqrt{26}$. This is wrong. Let's use another approach.
Let's use the ratio of corresponding side - lengths by counting grid - units.
The length of BC (by counting grid - units for right - triangle formed with horizontal and vertical segments) has a horizontal run of 2 units and vertical rise of 1 unit.
The length of B'C' has a horizontal run of 5 units and vertical rise of 1 unit.
If we consider the ratio of the lengths of corresponding sides, say comparing the horizontal lengths of BC and B'C' (or we can use other corresponding sides). The length of the horizontal part of BC is 2 units and of B'C' is 5 units.
The dilation factor $k$ is found by taking the ratio of the length of a side of the dilated triangle to the length of the corresponding side of the original triangle.
Let's take the distance from B to C horizontally (2 units) and from B' to C' horizontally (5 units). The dilation factor $k=\frac{\text{length of side in dilated figure}}{\text{length of corresponding side in original figure}}$.
If we count the grid - units for corresponding sides, for example, if we consider the horizontal distance from B to C and B' to C', the original horizontal distance from B to C is 2 units and from B' to C' is 5 units. But a simpler way is to note that if we consider the distance from the origin to a point on the triangle and its dilated counterpart.
Let's consider point B(-1,2) and B'(-2,4). The coordinates of B' are obtained from B by multiplying each coordinate by 2.
The dilation factor is 2.

Answer:

2