Question

Critical Thinking \(A B C D\) is a quadrilateral. \(E\) is the midpoint of \(\overline{A D}, F\) is the midpoint of \(\overline{D C}, G\) is the midpoint of \(\overline{A B}\), and \(H\) is the midpoint of \(\overline{B C}\). a. What can you say about \(\overline{E F}\) and \(\overline{G H}\) ? Explain. (Hint: Draw diagonal \(\overline{A C}\).) b. What kind of figure is EFHG?

Short answer

\(\overline{EF} \parallel \overline{GH}\) and \(EFHG\) is a parallelogram.

Step-by-step solution

Introduce the Midpoints

To solve this problem, identify the midpoints. Given in the quadrilateral \(ABCD\), \(E\) is the midpoint of \(\overline{AD}\), \(F\) is the midpoint of \(\overline{DC}\), \(G\) is the midpoint of \(\overline{AB}\), and \(H\) is the midpoint of \(\overline{BC}\).

Draw Diagonal AC

Draw the diagonal \(\overline{AC}\) as suggested by the hint. This diagonal divides the quadrilateral \(ABCD\) into two triangles, \(\triangle ABC\) and \(\triangle ADC\).

Analyze Midpoints in Triangles

In \(\triangle ADC\), since \(E\) and \(F\) are midpoints of \(\overline{AD}\) and \(\overline{DC}\) respectively, \(\overline{EF}\) is parallel to \(\overline{AC}\) and is half its length (midsegment theorem). In \(\triangle ABC\), since \(G\) and \(H\) are midpoints of \(\overline{AB}\) and \(\overline{BC}\) respectively, \(\overline{GH}\) is also parallel to \(\overline{AC}\) and half its length.

Conclude about Parallels and Lengths

Since both \(\overline{EF}\) and \(\overline{GH}\) are parallel to \(\overline{AC}\) and equal to half the length of \(\overline{AC}\), it follows that \(\overline{EF} \parallel \overline{GH}\) and \(\overline{EF} = \overline{GH}\).

Determine Figure EFHG

Since both pairs of opposite sides of quadrilateral \(EFHG\) are parallel and equal in length, \(EFHG\) must be a parallelogram.

Key concepts

Midpoints

The concept of a midpoint is fundamental in geometry. In a line segment, the midpoint is the point that divides the segment into two equal parts. Imagine a line segment, like a simple stick; a midpoint is simply its center. In our exercise, we identify several midpoints:

• Point E is the midpoint of segment \(\overline{AD}\).
• Point F is the midpoint of segment \(\overline{DC}\).
• Point G is the midpoint of segment \(\overline{AB}\).
• Point H is the midpoint of segment \(\overline{BC}\).

Understand that by finding the midpoint, you are creating two equal segments. For instance, if \(A\) to \(D\) is 10 units, \(A\) to \(E\) and \(E\) to \(D\) are both 5 units each. Midpoints are also important as they often help in dividing shapes into smaller, manageable parts.

Quadrilateral

A quadrilateral is a polygon with four edges (or sides) and four vertices (or corners). Think of them as the realized groceries list of geometry; they appear everywhere, from squares to rectangles to trapezoids. In our scenario:

• \(ABCD\) is the quadrilateral in question.

Quadrilaterals are versatile, and their properties change based on their types. Some quadrilaterals, like squares, have all sides equal, others like rectangles, have opposite sides equal. Understanding these properties helps in analyzing problems where they occur. The diagonals in quadrilaterals, such as \( \overline{AC} \), are often drawn to create triangles and simplify complicated shapes.

Parallel Lines

Parallel lines are like train tracks; they never, ever meet, no matter how far you extend them. In the exercise, knowing that certain line segments are parallel is central to the solution.In both triangles \( \triangle ABC \) and \( \triangle ADC \), we used the midsegment theorem which tells us:

• If a line segment joins the midpoints of two sides of a triangle, it is parallel to the third side.
• This line is also half the length of the third side.

Applying this creates segments \( \overline{EF} \) and \( \overline{GH} \) that are parallel to the diagonal \( \overline{AC} \) and equal in length. This affirmation is powerful, offering consistency in the logical structure of geometry.

Parallelogram

A parallelogram is a special type of quadrilateral with some special powers: opposite sides are parallel and of equal length. In the end, the understanding of parallelograms connects the dots in the exercise.Concluding the exercise, we found that \(EFHG\) forms a parallelogram because:

• \( \overline{EF} \parallel \overline{GH} \) and \( \overline{EG} = \overline{FH} \)
• Both of these pairs of opposite sides are equal and parallel.

This makes \(EFHG\) not just any quadrilateral, but specifically a parallelogram. Familiarity with shapes, especially parallelograms, not only bolsters problem-solving but also enriches understanding of how various geometric concepts interweave to form a coherent structure.