Coordinate Representation Math Example 4

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Example 4

hard
Three vertices of a parallelogram are A(1,2)A(1,2), B(5,2)B(5,2), C(7,6)C(7,6). Using coordinate methods, find the fourth vertex DD and compute the area of the parallelogram.

Solution

  1. 1
    Step 1: In parallelogram ABCDABCD, AB→=DC→\overrightarrow{AB} = \overrightarrow{DC}. AB→=(4,0)\overrightarrow{AB} = (4,0), so D=C−(4,0)=(3,6)D = C - (4,0) = (3, 6).
  2. 2
    Step 2: Base ∣AB∣=4|AB| = 4. Height == vertical distance from ABAB (along y=2y=2) to CC or DD: ∣6−2∣=4|6-2| = 4.
  3. 3
    Step 3: Area =base×height=4×4=16= \text{base} \times \text{height} = 4 \times 4 = 16 square units.

Answer

D(3,6)D(3, 6); area =16= 16 square units.
Coordinate geometry makes abstract geometric relationships computational. The fourth vertex is found using vector addition, and the area is computed from base and height read from the coordinates.

About Coordinate Representation

Describing geometric objects precisely using ordered pairs (x,y)(x, y) or triples (x,y,z)(x, y, z) in a coordinate system.

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More Coordinate Representation Examples

Example 1 easy

Plot points [formula], [formula], [formula] on the coordinate plane and find the distance from [form

Example 2 medium

Write the equation of the circle with centre [formula] and radius [formula] in standard form. Verify

Example 3 easy

Find the midpoint of the segment joining [formula] and [formula].

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