Question

the triangle jkl is a dilation of the triangle jkl. what is the scale factor of the dilation? simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.

Explanation:

Step1: Find coordinates of original and dilated points

First, identify coordinates of points in triangle \(JKL\) and \(J'K'L'\). Let's take point \(J\), \(K\), \(L\) and their dilated counterparts \(J'\), \(K'\), \(L'\).
- For \(J\): From the graph, \(J\) is at \((-1, -2)\)? Wait, no, looking at the grid: \(J\) is at \((-1, -2)\)? Wait, actually, let's check the grid lines. Wait, \(J\) is at \(x=-1\), \(y=-2\)? Wait, no, the green points: \(J\) is at \((-1, -2)\)? Wait, \(L\) is at \((-1, 2)\), \(J\) at \((-1, -2)\), \(K\) at \((1, -2)\). Then \(J'\) is at \((-5, -10)\), \(K'\) at \((5, -10)\), \(L'\) at \((-5, 10)\).

Step2: Calculate the scale factor

Scale factor \(k\) is the ratio of the length of a side in the dilated figure to the corresponding side in the original figure. Let's take the vertical side \(JL\) and \(J'L'\).
- Length of \(JL\): Distance between \(J(-1, -2)\) and \(L(-1, 2)\) is \(|2 - (-2)| = 4\) (since \(x\)-coordinates are same, distance is difference in \(y\)-coordinates).
- Length of \(J'L'\): Distance between \(J'(-5, -10)\) and \(L'(-5, 10)\) is \(|10 - (-10)| = 20\).
- Scale factor \(k=\frac{\text{Length of }J'L'}{\text{Length of }JL}=\frac{20}{4}=5\)? Wait, no, wait, maybe I got the coordinates wrong. Wait, let's recheck the grid. Wait, the original triangle \(JKL\): \(J\) is at \((-1, -2)\)? Wait, no, the grid has each square as 1 unit. Let's look at the green points: \(J\) is at \((-1, -2)\), \(L\) at \((-1, 2)\), \(K\) at \((1, -2)\). Then \(J'\) is at \((-5, -10)\), \(K'\) at \((5, -10)\), \(L'\) at \((-5, 10)\). Wait, but maybe the original triangle is smaller. Wait, another way: the vector from \(J\) to \(J'\): \(J\) is at \((-1, -2)\), \(J'\) at \((-5, -10)\). The change in \(x\) is \(-5 - (-1)= -4\), change in \(y\) is \(-10 - (-2)= -8\). Wait, but dilation from the origin? Wait, maybe the center of dilation is the origin. Let's check the coordinates:

Wait, maybe the original points are \(J(-1, -2)\), \(K(1, -2)\), \(L(-1, 2)\). Then dilated points: \(J'(-5, -10)\), \(K'(5, -10)\), \(L'(-5, 10)\). So the coordinates of \(J'\) are \(5 \times (-1, -2)\), \(K'\) are \(5 \times (1, -2)\), \(L'\) are \(5 \times (-1, 2)\). So the scale factor is 5? Wait, no, wait, maybe I messed up the original coordinates. Wait, let's check the grid again. The original triangle \(JKL\): \(J\) is at \((-1, -2)\)? Wait, the green segment \(JL\) is from \(y=-2\) to \(y=2\), so length 4. The dilated segment \(J'L'\) is from \(y=-10\) to \(y=10\), length 20. So \(20/4 = 5\)? Wait, but that seems too big. Wait, maybe the original points are \(J(-1, -2)\), \(K(1, -2)\), \(L(-1, 2)\), and dilated points are \(J'(-5, -10)\), \(K'(5, -10)\), \(L'(-5, 10)\). So the scale factor is \(10/2 = 5\)? Wait, no, the vertical distance from \(J\) to \(L\) is \(2 - (-2) = 4\), and from \(J'\) to \(L'\) is \(10 - (-10) = 20\), so \(20/4 = 5\). Alternatively, the horizontal distance from \(J\) to \(K\) is \(1 - (-1) = 2\), and from \(J'\) to \(K'\) is \(5 - (-5) = 10\), so \(10/2 = 5\). So scale factor is 5? Wait, but maybe I made a mistake. Wait, let's check the coordinates again. Wait, the original triangle: \(J\) is at \((-1, -2)\), \(K\) at \((1, -2)\), \(L\) at \((-1, 2)\). So it's a right triangle with legs 2 (horizontal) and 4 (vertical). The dilated triangle: \(J'\) at \((-5, -10)\), \(K'\) at \((5, -10)\), \(L'\) at \((-5, 10)\). So horizontal leg: \(5 - (-5) = 10\), vertical leg: \(10 - (-10) = 20\). So scale factor is \(10/2 = 5\) or \(20/4 = 5\). So the scale factor is 5? Wait, but maybe the original triangle is smaller. Wait, maybe the original points are \(J(-1, -2)\), \(K(1, -2)\), \(L(-1, 2)\), and the dilated points are \(J'(-5, -10)\), \(K'(5, -10)\), \(L'(-5, 10)\). So the scale factor is \(5\), because each coordinate is multiplied by 5: \(-1 \times 5 = -5\), \(-2 \times 5 = -10\), etc. So scale factor is 5.

Answer:

\(5\)