Question

in the following triangle, point e is the midpoint of \\(\overline{ab}\\), and point d is the midpoint of \\(\overline{ac}\\).

(image of triangle with points c, b, a, d, e)

below is the proof that \\(de = \frac{1}{2}cb\\). the proof is divided into four parts, where the title of each part indicates its main purpose.

complete part d of the proof.

part a: prove \\(\frac{ca}{da} = 2\\)
show the steps.

part b: prove \\(\frac{ba}{ea} = 2\\)
show the steps.

part c: prove \\(\triangle cab \sim \triangle dae\\)
show the steps.

part d: prove \\(de = \frac{1}{2}cb\\)

12 \\(\frac{cb}{de} = \\) pick ratio reason: lengths of corresponding sides of similar triangles have equal ratios. (part c)
13 \\(\frac{cb}{de} = \\) blank reason: substitution (part a, 12)
14 \\(\frac{1}{2}cb = de\\) reason: multiply both sides of the equation by pick expression (13)

Explanation:

Step1: Determine the ratio for similar triangles

Since \(\triangle CAB \sim \triangle DAE\) (from Part C), the ratio of corresponding sides \(\frac{CB}{DE}\) should be equal to the ratio of other corresponding sides. From Part A, \(\frac{CA}{DA} = 2\), and for similar triangles, corresponding sides are proportional, so \(\frac{CB}{DE}=\frac{CA}{DA} = 2\). So the ratio in step 12 is \(2\).

Step2: Substitute the ratio

In step 13, using substitution from Part A (where \(\frac{CA}{DA} = 2\)) and the proportionality from similar triangles, we substitute into \(\frac{CB}{DE}\), so \(\frac{CB}{DE}=2\).

Step3: Solve for \(DE\)

To get \(DE=\frac{1}{2}CB\), we multiply both sides of the equation \(\frac{CB}{DE}=2\) by \(\frac{DE}{2}\) (or equivalently, divide both sides by \(2\) and multiply by \(DE\)). So we multiply both sides by \(\frac{DE}{2}\), which gives \(\frac{1}{2}CB = DE\).

Answer:

• Step 12: The ratio is \(2\), so \(\frac{CB}{DE} = 2\)
• Step 13: Substituting, we get \(\frac{CB}{DE}=2\)
• Step 14: Multiply both sides by \(\frac{DE}{2}\) (or equivalent expression to isolate \(DE\)) to get \(\frac{1}{2}CB = DE\)

(For the "Pick ratio" in step 12, choose \(2\); for step 13, the value is \(2\); for "Pick expression" in step 14, the expression is \(\frac{DE}{2}\) (or similar to multiply both sides to solve for \(DE\)))