On the coordinate plane, the segment from U(4,–6) to V(–8,–11) forms one side of a rectangle. The rectangle has a perimeter of 78 units. Find the area of the rectangle.
Question
On the coordinate plane, the segment from U(4,–6) to V(–8,–11) forms one side of a rectangle. The rectangle has a perimeter of 78 units. Find the area of the rectangle.
Solution
Step 1: Find the length of the segment UV
The length of a line segment with endpoints (x1, y1) and (x2, y2) is given by the formula √[(x2 - x1)² + (y2 - y1)²].
So, the length of UV = √[(-8 - 4)² + (-11 - (-6))²] = √[(-12)² + (-5)²] = √[144 + 25] = √169 = 13 units.
Step 2: Find the length of the other side of the rectangle
The perimeter of a rectangle is given by the formula 2(length + width). Given that the perimeter is 78 units and one side (UV) is 13 units, we can set up the equation 2(13 + width) = 78. Solving for width gives us width = 26 units.
Step 3: Find the area of the rectangle
The area of a rectangle is given by the formula length * width. So, the area of the rectangle is 13 units * 26 units = 338 square units.
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