4.1 Q-11

Question Statement

Find the value of $h$ such that the quadrilateral with vertices $A(-3,0)$, $B(1,-2)$, $C(5,0)$, and $D(1,h)$ forms a parallelogram. Additionally, determine if the quadrilateral is a square.

---

Background and Explanation

To solve this problem, we need to use the properties of a parallelogram and a square:

• In a parallelogram, opposite sides are equal in length.
• A square is a special type of parallelogram where all four sides are equal in length, and the angles between adjacent sides are 90°.

We will first use the condition for a parallelogram (equal opposite sides) to find $h$, and then check if the quadrilateral is a square.

---

Solution

Step 1: Use the condition for a parallelogram

For the quadrilateral $A B C D$ to be a parallelogram, the lengths of opposite sides must be equal. Specifically, we need to ensure that:

$$|AB| = |CD| \quad \text{and} \quad |BC| = |AD|$$

---

Step 2: Find the length of side $AB$

The distance between points $A(-3, 0)$ and $B(1, -2)$ can be calculated using the distance formula:

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

Substitute the coordinates of points $A$ and $B$:

$$|AB| = \sqrt{(1 - (-3))^2 + (-2 - 0)^2} = \sqrt{(4)^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20}$$

So, the length of side $AB$ is $\sqrt{20}$.

---

Step 3: Find the length of side $CD$

Next, we find the length of side $CD$ using the coordinates of points $C(5, 0)$ and $D(1, h)$:

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

Substitute the coordinates of points $C$ and $D$:

$$|CD| = \sqrt{(1 - 5)^2 + (h - 0)^2} = \sqrt{(-4)^2 + h^2} = \sqrt{16 + h^2}$$

Since $|AB| = |CD|$, we set the two expressions equal to each other:

$$\sqrt{20} = \sqrt{16 + h^2}$$

---

Step 4: Solve for $h$

Square both sides of the equation to eliminate the square roots:

$$20 = 16 + h^2$$

Subtract 16 from both sides:

$$4 = h^2$$

Take the square root of both sides:

$$h = \pm 2$$

Thus, $h$ can be either $2$ or $-2$.

---

Step 5: Check if the quadrilateral is a square

For the quadrilateral to be a square, not only must the opposite sides be equal, but all four sides must also be of equal length. We check if the side lengths $AC$ and $BD$ are equal when $h = 2$.

Length of $AC$:

The distance between points $A(-3, 0)$ and $C(5, 0)$ is:

$$|AC| = \sqrt{(5 - (-3))^2 + (0 - 0)^2} = \sqrt{(8)^2} = 8$$

Length of $BD$:

The distance between points $B(1, -2)$ and $D(1, 2)$ (when $h = 2$) is:

$$|BD| = \sqrt{(1 - 1)^2 + (2 - (-2))^2} = \sqrt{(0)^2 + (4)^2} = 4$$

Since $|AC| \neq |BD|$, the quadrilateral is not a square.

---

Key Formulas or Methods Used

• Distance Formula:

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

• Condition for a Parallelogram: Opposite sides must be equal in length.

---

Summary of Steps

1. Find the length of side $AB$ using the distance formula: $|AB| = \sqrt{20}$.
2. Find the length of side $CD$: $|CD| = \sqrt{16 + h^2}$.
3. Set $|AB| = |CD|$ and solve for $h$: $h = \pm 2$.
4. Check if the quadrilateral is a square by comparing $AC$ and $BD$.
5. Since $AC \neq BD$, the quadrilateral is not a square.