Question

δjkl was dilated by a factor of 2/3 to form δjkl. find x and y.
mber: dilations form similar triangles!
(there are two triangles in the image, one labeled with k, l, and a side of 13.5, angle 43° at l, and another labeled k, l, j with side 12, x, and angle y° at l)

Explanation:

Step1: Find the scale factor relation

Dilation forms similar triangles, so corresponding sides are proportional, and corresponding angles are equal. Let's first find the length of \( JK \) (assuming the side with length 8 in \( \triangle JKL \) is \( JK \), and \( J'K' = 12 \)). The scale factor \( k=\frac{2}{3} \), but wait, \( \frac{J'K'}{JK}=\frac{12}{JK}=\frac{2}{3} \)? Wait, no, maybe I mixed up. Wait, the dilation factor is \( \frac{2}{3} \), so \( \text{Length of image} = \text{Length of original} \times \text{dilation factor} \). Wait, let's check the sides. In \( \triangle JKL \), let's say \( JK = 18 \) (since the left side has 8? Wait, the image is a bit unclear, but the key is that for similar triangles, angles are equal, so \( y = 43^\circ \). For the side \( x \): in \( \triangle JKL \), \( KL = 13.5 \), so \( x = KL \times \frac{2}{3} \).

Step2: Calculate x

\( x = 13.5 \times \frac{2}{3} \). Let's compute that: \( 13.5 \div 3 = 4.5 \), \( 4.5 \times 2 = 9 \). So \( x = 9 \).

Step3: Determine y

Since dilation preserves angle measures (similar triangles have equal corresponding angles), \( \angle L' = \angle L \), so \( y = 43 \).

Answer:

\( x = 9 \), \( y = 43^\circ \)