To find the value of '\(a\)' for the parallelogram with vertices \(A(1, 2)\), \(B(a, -26)\), \(C(13, -14)\), and \(D(6, 14)\) taken in order, we use the property that the diagonals of a parallelogram bisect each other.
Parallelogram Vertices Property
The midpoint of diagonal \(AC\) must be the same as the midpoint of diagonal \(BD\). The midpoint formula for two points \((x_1, y_1)\) and \((x_2, y_2)\) is \((\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})\).
Calculate Diagonal Midpoints
- Midpoint of AC: Using points \(A(1, 2)\) and \(C(13, -14)\):
Midpoint\(_{AC} = (\frac{1+13}{2}, \frac{2+(-14)}{2}) = (\frac{14}{2}, \frac{-12}{2}) = (7, -6)\)
- Midpoint of BD: Using points \(B(a, -26)\) and \(D(6, 14)\):
Midpoint\(_{BD} = (\frac{a+6}{2}, \frac{-26+14}{2}) = (\frac{a+6}{2}, \frac{-12}{2}) = (\frac{a+6}{2}, -6)\)
Solve for Unknown Coordinate 'a'
Equating the x-coordinates of the midpoints:
\(7 = \frac{a+6}{2}\)
Multiply both sides by 2:
\(7 \times 2 = a+6\)
\(14 = a+6\)
Subtract 6 from both sides:
\(a = 14 - 6\)
\(a = 8\)
Thus, the value of '\(a\)' is 8.
Similar Questions
- What would be the area of the triangle with \(A(8, 8)\), \(B(22, 8)\) and \(C(16, -10)\) as vertices?
- What would be the point which divides the line segment connecting the points \(P(10, 18), Q(5, 8)\) in the ratio \(2 : 3\) internally
- Find a point on the X axis which is equidistant from A(2, -3) and B(-2, 1)
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Important Questions from Coordinate Geometry
Find the co-ordinates of the centroid of a triangle whose vertices are A(1, 4), B(7, 8) and C(10, 12).
- If x² + y² - 16x + 38y + 425 = 0, then the value of x² + y² is:
- If x² + y² - 12x + 18y + 117 = 0, then the value of x² + y² is:
- A geological survey team has established three research stations forming a triangular monitoring zone. The stations are located at coordinates P(2,5), Q(8,1), and R(4,9) on a coordinate grid where each unit represents 1 kilometer. What is the area of the triangular monitoring zone?
- What would be the area of the triangle with \(A(8, 8)\), \(B(22, 8)\) and \(C(16, -10)\) as vertices?
