Find the area of a rhombus if its vertices are (3, 0), (4, 5), (-1, 4) and (-2, -1) taken in order. [Hint: Area of a rhombus = 1/2 (product of its diagonals)]
Let the vertices of the rhombus be A(3, 0), B(4, 5), C(-1, 4) and D(-2, -1) taken in order. The diagonals of the rhombus are AC and BD.
Length of diagonal AC: Using the distance formula, AC = √[(-1 - 3)² + (4 - 0)²] AC = √[(-4)² + (4)²] = = = 4 units.
Length of diagonal BD: Using the distance formula, BD = √[(-2 - 4)² + (-1 - 5)²] BD = √[(-6)² + (-6)²] = = = 6 units.
Area of the rhombus: Area = 1/2 × (Product of diagonals) Area = 1/2 × (4√2 \times 6$$$\sqrt{2}$) = 1/2 × (24 \times 2$$) = 24 sq. units.
Explanation
The problem requires finding the area of a rhombus given its vertices. The provided hint specifies that the area is half the product of its diagonals. The context from Example 2 illustrates how to calculate the distance between two points (diagonals in this case) using the distance formula. By identifying the vertices in order, we determine the diagonals are AC and BD. Calculating these lengths allows us to apply the area formula directly.
Solution Steps
Step 1: Identify the vertices in order as A(3, 0), B(4, 5), C(-1, 4), and D(-2, -1). The diagonals are AC and BD.
Step 2: Calculate the length of diagonal AC using the distance formula: AC = √[(-1 - 3)² + (4 - 0)²] = = = 4.
Step 3: Calculate the length of diagonal BD using the distance formula: BD = √[(-2 - 4)² + (-1 - 5)²] = = = 6.
Step 4: Calculate the area using the formula: Area = 1/2 × AC × BD = 1/√$$2 \times 6$$$\sqrt{2}$ = 24 sq. units.