Question

question
triangle fgh is formed by connecting the midpoints of the side of triangle cde. the lengths of the sides of triangle cde are shown. what is the length of (overline{hg})? figures not necessarily drawn to scale.

Explanation:

Step1: Recall Midsegment Theorem

The Midsegment Theorem (also known as the Midline Theorem) states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long.

Step2: Identify the triangle and midpoints

In triangle \( CDE \), \( F \), \( G \), and \( H \) are midpoints (since \( FGH \) is formed by connecting midpoints). We need to find \( HG \). Let's see the side it's related to. Looking at the triangle, \( HG \) should be the midsegment of triangle \( CDE \) with respect to side \( CD \) (or another side, but let's check the lengths). The sides of \( CDE \) are each 8? Wait, the diagram shows \( CD = 8 \), \( DE = 8 \), \( CE = 8 \)? Wait, no, the labels: \( CH \) and \( HE \) are parts of \( CE \), \( DG \) and \( GE \) parts of \( DE \), \( CF \) and \( FD \) parts of \( CD \), each segment (like \( CH \) to \( HE \))? Wait, no, \( F \), \( G \), \( H \) are midpoints, so \( H \) is midpoint of \( CE \), \( G \) midpoint of \( DE \), \( F \) midpoint of \( CD \). Then \( HG \) connects midpoints of \( CE \) and \( DE \), so it should be midsegment to \( CD \).

Step3: Apply Midsegment Theorem

The length of the midsegment is half the length of the third side. The third side here is \( CD \), which has length 8. So \( HG=\frac{1}{2}\times CD \).

Step4: Calculate \( HG \)

Substitute \( CD = 8 \) into the formula: \( HG=\frac{1}{2}\times8 = 4 \).

Answer:

\( 4 \)