the vertices (corners) of a rectangle are A(1,3), B(1,-1) and C(7,-1). plot these points on a graph paper and hence use it to find the coordinates of the fourth vertex. also find the area of the rectangle as well as the point of intersection of diagonals from the graph
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To plot the given points A(1,3), B(1,-1), and C(7,-1) on a graph paper, follow these steps:
1. Draw the X and Y axes on the graph paper.
2. Locate the point A(1,3) by moving 1 unit to the right along the X-axis and 3 units upwards along the Y-axis. Mark this point as A.
3. Locate the point B(1,-1) by moving 1 unit to the right along the X-axis and 1 unit downwards along the Y-axis. Mark this point as B.
4. Locate the point C(7,-1) by moving 7 units to the right along the X-axis and staying at the same level along the Y-axis. Mark this point as C.
5. Connect the points A, B, C to form a rectangle.
The graph should now display the rectangle with vertices A, B, and C.
To find the coordinates of the fourth vertex, let's denote the fourth vertex as D(x, y).
Since a rectangle has opposite sides parallel and equal in length, we can find the length of AB and BC:
AB = Distance between A and B = √[(x₂ - x₁)² + (y₂ - y₁)²]
= √[(1 - 1)² + (-1 - 3)²]
= √[0² + (-4)²]
= √(0 + 16)
= 4
BC = Distance between B and C = √[(x₂ - x₁)² + (y₂ - y₁)²]
= √[(7 - 1)² + (-1 - (-1))²]
= √[6² + 0²]
= √36
= 6
Since AB = BC, the rectangle is a square, and the length of all sides is 4.
To find the coordinates of the fourth vertex, we can use the distance formula:
BD = √[(x₂ - x₁)² + (y₂ - y₁)²]
4 = √[(x - 1)² + (y - (-1))²]
16 = (x - 1)² + (y + 1)²
We also know that AD and DC are vertical lines, so the X-coordinate of D will be the same as that of A and C.
Therefore, D(x, y) has the coordinates D(7, 3).
To find the area of the rectangle, we can use the formula:
Area = Length × Width
Area = AB × BC
Area = 4 × 6
Area = 24 square units
The point of intersection of the diagonals can be found by calculating the midpoint of the line segment AC, as the diagonals of a rectangle bisect each other.
Midpoint of AC = ( (x₁ + x₂) / 2, (y₁ + y₂) / 2 )
Midpoint of AC = ( (1 + 7) / 2, (3 + (-1)) / 2 )
Midpoint of AC = ( 8 / 2, 2 / 2 )
Midpoint of AC = ( 4, 1 )
Therefore, the point of intersection of the diagonals is (4, 1).
Note: The above calculations are based on the assumption that the given points indeed form a rectangle.
Answer:
yes provided on graph paper can help it
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