Question

1. in △abc, d and e are midpoints of ab and ac. if ab = 12 and ac = 16, what is the length of de? a. 6 b. 10 c. 9 d. 8

Explanation:

Step1: Apply the mid - segment theorem

The mid - segment theorem states that the segment connecting the mid - points of two sides of a triangle is parallel to the third side and half its length. In \(\triangle ABC\), since \(D\) is the mid - point of \(AB\) and \(E\) is the mid - point of \(AC\), \(DE\) is the mid - segment.

Step2: Calculate the length of \(BC\) using the Pythagorean theorem (assuming \(\triangle ABC\) is a right - triangle, if not, we can also use the formula for the distance between two points in coordinate geometry or vector methods. But if we assume it's a right - triangle with \(AB = 12\) and \(AC=16\))

By the Pythagorean theorem \(BC=\sqrt{AB^{2}+AC^{2}}=\sqrt{12^{2}+16^{2}}=\sqrt{144 + 256}=\sqrt{400}=20\)

Step3: Use the mid - segment formula

Since \(DE=\frac{1}{2}BC\) (mid - segment theorem), substituting \(BC = 20\) into the formula, we get \(DE=\frac{1}{2}\times20 = 10\)

Answer:

b. 10