Question

1. given that segment de is a mid - segment of δabc, find the following.

ad =
de =
bc =
5.4

Explanation:

Step1: Recall mid - segment property

A mid - segment of a triangle is parallel to the third side and half its length. Also, a mid - segment divides the two sides it intersects into equal parts.

Step2: Find the length of AD

Since DE is a mid - segment of $\triangle ABC$, D is the mid - point of AB. Given $AB = 5$, then $AD=\frac{AB}{2}=\frac{5}{2}=2.5$.

Step3: Find the length of DE

The mid - segment DE is half the length of the side BC that it is parallel to. First, we find BC using the fact that E is the mid - point of AC. Given $AC = 12.6$, E is the mid - point of AC. And since DE is parallel to BC, by the mid - segment theorem, $DE=\frac{BC}{2}$. Also, since E is the mid - point of AC, we know that the ratio of the sides of similar triangles $\triangle ADE$ and $\triangle ABC$ (by AA similarity as $\angle A$ is common and $\angle ADE=\angle ABC$ because $DE\parallel BC$) gives us the relationship. Since E is the mid - point of AC, $AC = 12.6$, and D is the mid - point of AB. The mid - segment $DE=\frac{BC}{2}$. We know that $BC = 4.1\times2 = 8.2$ (because of the mid - segment property), so $DE=\frac{BC}{2}=\frac{8.2}{2}=4.1$.

Step4: Find the length of BC

As mentioned above, since DE is a mid - segment of $\triangle ABC$, the side BC is twice the length of the mid - segment DE. So $BC = 8.2$.

Answer:

$AD = 2.5$
$DE = 4.1$
$BC = 8.2$