4.1 Q-10

Question Statement

Given the quadrilateral with vertices $A(9, 3)$, $B(-7, 7)$, $C(-3, -7)$, and $D(5, -5)$, find the midpoints of its sides. Show that the figure formed by joining the midpoints consecutively is a parallelogram.

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Background and Explanation

To solve this problem, we need to recall the concept of the midpoint of a line segment. The midpoint of a segment connecting two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:

$$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

We will find the midpoints of the sides of the quadrilateral formed by points $A$, $B$, $C$, and $D$. Then, we will verify that the figure formed by joining these midpoints is a parallelogram. For a quadrilateral to be a parallelogram, the opposite sides must be equal in length.

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Solution

Step 1: Find the midpoints of the sides

We will apply the midpoint formula to each side of the quadrilateral.

• Midpoint of $\overline{AB}$:
  The coordinates of $A$ are $(9, 3)$ and $B$ are $(-7, 7)$. Using the midpoint formula:

$$D = \left(\frac{9 + (-7)}{2}, \frac{3 + 7}{2}\right) = D(1, 5)$$

• Midpoint of $\overline{BC}$:
  The coordinates of $B$ are $(-7, 7)$ and $C$ are $(-3, -7)$. Using the midpoint formula:

$$E = \left(\frac{-7 + (-3)}{2}, \frac{7 + (-7)}{2}\right) = E(-5, 0)$$

• Midpoint of $\overline{CD}$:
  The coordinates of $C$ are $(-3, -7)$ and $D$ are $(5, -5)$. Using the midpoint formula:

$$F = \left(\frac{-3 + 5}{2}, \frac{-7 + (-5)}{2}\right) = F(1, -6)$$

• Midpoint of $\overline{DA}$:
  The coordinates of $D$ are $(5, -5)$ and $A$ are $(9, 3)$. Using the midpoint formula:

$$G = \left(\frac{9 + 5}{2}, \frac{3 + (-5)}{2}\right) = G(7, -1)$$

Step 2: Prove that the figure formed by the midpoints is a parallelogram

To prove that the quadrilateral $DEFG$ is a parallelogram, we need to show that opposite sides are equal. We will calculate the lengths of the sides of quadrilateral $DEFG$.

• Length of $DE$:
  The coordinates of $D$ are $(1, 5)$ and $E$ are $(-5, 0)$. Using the distance formula:

$$|DE| = \sqrt{(-5 - 1)^2 + (0 - 5)^2} = \sqrt{(-6)^2 + (-5)^2} = \sqrt{36 + 25} = \sqrt{61}$$

• Length of $FG$:
  The coordinates of $F$ are $(1, -6)$ and $G$ are $(7, -1)$. Using the distance formula:

$$|FG| = \sqrt{(7 - 1)^2 + (-1 - (-6))^2} = \sqrt{6^2 + 5^2} = \sqrt{36 + 25} = \sqrt{61}$$

• Length of $EF$:
  The coordinates of $E$ are $(-5, 0)$ and $F$ are $(1, -6)$. Using the distance formula:

$$|EF| = \sqrt{(1 - (-5))^2 + (-6 - 0)^2} = \sqrt{6^2 + (-6)^2} = \sqrt{36 + 36} = \sqrt{72}$$

• Length of $GD$:
  The coordinates of $G$ are $(7, -1)$ and $D$ are $(1, 5)$. Using the distance formula:

$$|GD| = \sqrt{(7 - 1)^2 + (-1 - 5)^2} = \sqrt{6^2 + (-6)^2} = \sqrt{36 + 36} = \sqrt{72}$$

Step 3: Conclude that the figure is a parallelogram

We have found that:

• $|DE| = |FG| = \sqrt{61}$
• $|EF| = |GD| = \sqrt{72}$

Since opposite sides are equal in length, quadrilateral $DEFG$ is a parallelogram.

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Key Formulas or Methods Used

• Midpoint Formula:
  The midpoint of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is:

$$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

• Distance Formula:
  The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

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Summary of Steps

1. Calculate the midpoints of the sides of the quadrilateral using the midpoint formula.
2. Use the distance formula to find the lengths of the sides of quadrilateral $DEFG$.
3. Check that opposite sides of $DEFG$ are equal in length.
4. Conclude that quadrilateral $DEFG$ is a parallelogram because opposite sides are equal.

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