Question

in the figure below, △edc is dilated to create △abc.
select options below

1. the scale factor in the dilation is
2. the center of dilation is
3. the length of (overline{ae}) is

Explanation:

Step1: Identify Similar Triangles

Since \(\triangle EDC\) is dilated to \(\triangle ABC\), they are similar. So, corresponding sides are proportional. The base of \(\triangle ABC\) is \(12\) cm, and the base of \(\triangle EDC\) is \(8\) cm. The height of \(\triangle ABC\) is \(AE + EC = AE + 10\) cm, and the height of \(\triangle EDC\) is \(EC = 10\) cm? Wait, no, let's check the diagram. Wait, \(AC\) is \(10\) cm? Wait, the vertical side: \(A\) to \(E\) to \(C\), with \(AE\) unknown, \(EC = 10\) cm? Wait, no, the horizontal sides: \(AB\) (wait, no, the top horizontal is \(12\) cm, the middle horizontal is \(8\) cm. The vertical side: \(A\) to \(C\) is \(10\) cm, with \(E\) between \(A\) and \(C\). So \(\triangle EDC \sim \triangle ABC\) by AA (right angles and common angle at \(C\)). So the ratio of sides is \(\frac{ED}{AB}=\frac{EC}{AC}\). Wait, \(ED = 8\) cm, \(AB = 12\) cm? Wait, no, \(ED\) is the base of \(\triangle EDC\), \(AB\) is the base of \(\triangle ABC\)? Wait, no, the diagram: \(A\) is top left, \(B\) top right, \(C\) bottom left, \(E\) on \(AC\), \(D\) on \(BC\). So \(AE\) is vertical segment, \(ED\) is horizontal, \(AB\) is horizontal (length \(12\) cm), \(ED\) is horizontal (length \(8\) cm), \(AC\) is vertical (length \(AE + EC = AE + 10\) cm), \(EC\) is vertical (length \(10\) cm)? Wait, no, \(AC\) is \(10\) cm? Wait, the label says \(10\) cm next to \(AC\). So \(AC = AE + EC = AE + 10\)? No, wait, \(E\) is between \(A\) and \(C\), so \(AC = AE + EC\), and \(EC = 10\) cm? Wait, no, the diagram shows \(A\) to \(E\) to \(C\), with \(EC = 10\) cm? Wait, maybe I misread. Wait, the vertical side: \(A\) is at the top, \(C\) at the bottom, so \(AC\) is vertical, length \(10\) cm? No, the label \(10\) cm is next to \(AC\), so \(AC = 10\) cm. Then \(E\) is a point on \(AC\), so \(AE + EC = AC = 10\) cm? No, that can't be. Wait, maybe the horizontal sides: \(AB\) (top) is \(12\) cm, \(ED\) (middle) is \(8\) cm. The vertical side: \(A\) to \(C\) is \(10\) cm, with \(E\) between \(A\) and \(C\), so \(AE\) is what we need to find. Since \(\triangle EDC \sim \triangle ABC\), the ratio of similarity is \(\frac{ED}{AB}=\frac{8}{12}=\frac{2}{3}\)? Wait, no, dilation: if \(\triangle EDC\) is dilated to \(\triangle ABC\), then the scale factor is \(\frac{AB}{ED}=\frac{12}{8}=\frac{3}{2}\). So the height of \(\triangle ABC\) (which is \(AC\)) should be \(\frac{3}{2}\) times the height of \(\triangle EDC\) (which is \(EC\)). Wait, \(EC = 10\) cm? Then \(AC = \frac{3}{2} \times EC\)? No, that would make \(AC = 15\) cm, but the diagram says \(10\) cm. Wait, maybe \(EC = 10\) cm, and \(AC = AE + EC\), so \(AE = AC - EC\)? Wait, no, maybe I got the triangles reversed. If \(\triangle EDC\) is dilated to \(\triangle ABC\), then \(\triangle ABC\) is larger, so scale factor is \(\frac{AB}{ED}=\frac{12}{8}=\frac{3}{2}\). Then the height of \(\triangle ABC\) (from \(A\) to \(C\)) is \(\frac{3}{2}\) times the height of \(\triangle EDC\) (from \(E\) to \(C\)). Let \(EC = h\), then \(AC = \frac{3}{2}h\). But \(AC = AE + h\), so \(AE + h = \frac{3}{2}h\), so \(AE = \frac{1}{2}h\). Wait, but the diagram labels \(AC\) as \(10\) cm? No, maybe the vertical side is \(AC = 10\) cm, so \(h = EC = 10\) cm? Then \(AC = \frac{3}{2} \times 10 = 15\) cm, which contradicts. Wait, maybe the vertical side is \(AE + EC = 10\) cm? No, this is confusing. Wait, let's look at the horizontal sides: \(AB = 12\) cm, \(ED = 8\) cm. The ratio of dilation is \(\frac{AB}{ED} = \frac{12}{8} = \frac{3}{2}\). Now, the vertical segments: \(AC\) is the height of \(\triangle ABC\), \(EC\) is the height of \(\triangle EDC\). So \(\frac{AC}{EC} = \frac{3}{2}\). Let \(EC = x\), then \(AC = \frac{3}{2}x\). But \(AC = AE + x\), so \(AE + x = \frac{3}{2}x\) \(\implies\) \(AE = \frac{1}{2}x\). Wait, but in the diagram, \(EC\) is labeled \(10\) cm? Wait, the label \(10\) cm is next to \(AC\), so \(AC = 10\) cm? Then \(\frac{10}{x} = \frac{3}{2}\) \(\implies\) \(x = \frac{20}{3}\), which is not nice. Wait, maybe I mixed up the triangles. Maybe \(\triangle ABC\) is dilated to \(\triangle EDC\), but the problem says \(\triangle EDC\) is dilated to \(\triangle ABC\), so \(\triangle ABC\) is the image, \(\triangle EDC\) is the pre-image. So scale factor \(k = \frac{\text{image side}}{\text{pre-image side}} = \frac{12}{8} = \frac{3}{2}\). Now, the height of the pre-image (\(\triangle EDC\)) is \(EC = 10\) cm? No, the height of the image (\(\triangle ABC\)) is \(AC = AE + EC = AE + 10\) cm, and the height of the pre-image (\(\triangle EDC\)) is \(EC = 10\) cm? No, that can't be. Wait, maybe the vertical side is \(AE = 10\) cm? No, the label \(10\) cm is next to \(AC\). Wait, maybe the diagram has \(AC = 10\) cm, \(AB = 12\) cm, \(ED = 8\) cm. Then, since \(\triangle EDC \sim \triangle ABC\), \(\frac{ED}{AB} = \frac{EC}{AC}\). So \(\frac{8}{12} = \frac{EC}{10}\) \(\implies\) \(EC = \frac{8 \times 10}{12} = \frac{20}{3}\), which is not nice. Wait, maybe I got the sides wrong. The horizontal sides: \(AB\) is \(12\) cm, \(ED\) is \(8\) cm. The vertical sides: \(AE\) is unknown, \(EC = 10\) cm. So \(AC = AE + 10\). Then \(\frac{ED}{AB} = \frac{EC}{AC}\) \(\implies\) \(\frac{8}{12} = \frac{10}{AE + 10}\) \(\implies\) \(8(AE + 10) = 12 \times 10\) \(\implies\) \(8AE + 80 = 120\) \(\implies\) \(8AE = 40\) \(\implies\) \(AE = 5\) cm. Ah, that makes sense! So the length of \(AE\) is \(5\) cm.

Step2: Verify the Scale Factor

The scale factor of dilation is \(\frac{AB}{ED} = \frac{12}{8} = \frac{3}{2}\). Let's check the vertical sides: \(AC = AE + EC = 5 + 10 = 15\) cm, \(EC = 10\) cm. \(\frac{AC}{EC} = \frac{15}{10} = \frac{3}{2}\), which matches the scale factor. So that's correct.

Step3: Find the Length of \(AE\)

From the equation above, we solved \(8(AE + 10) = 120\) \(\implies\) \(AE + 10 = 15\) \(\implies\) \(AE = 5\) cm.

Answer:

The length of \(\overline{AE}\) is \(\boxed{5}\) cm.