Locate the points A(1, 2), B(3, 4) and C(5, 2) on a graph sheet taking suitable axes. Write the coordinates of the fourth point D in order to complete the rhombus ABCD. Measure the diagonals of this rhombus and check whether they are equal or not.

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Plot the Points A, B, and C
- **Point A (1, 2)**: Start at the origin (0, 0). Move 1 unit to the right (along the x-axis) and 2 units up (along the y-axis). Mark this point as A.
- **Point B (3, 4)**: From the origin, move 3 units to the right and 4 units up. Mark this point as B.
- **Point C (5, 2)**: From the origin, move 5 units to the right and 2 units up. Mark this point as C.

### Step 2: Determine the Coordinates of Point D
To complete the rhombus ABCD, we need to find the coordinates of point D. A rhombus has equal diagonals, and the diagonals bisect each other at right angles.

- The midpoints of the diagonals will be the same. The midpoint of AC can be calculated as:
\[ \text{Midpoint of AC} = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2} \right) = \left( \frac{1 + 5}{2}, \frac{2 + 2}{2} \right) = (3, 2) \]

- The coordinates of point D can be determined using the fact that it must also be equidistant from the midpoint of AC. Since we have point B at (3, 4), point D must be directly below point B to maintain the rhombus shape. Thus, point D will be at (3, 0).

### Step 3: Plot Point D
- **Point D (3, 0)**: From the origin, move 3 units to the right and 0 units up (which means you stay on the x-axis). Mark this point as D.

### Step 4: Measure the Diagonals
Now we will measure the lengths of the diagonals AC and BD.

- **Diagonal AC**:
- The distance between points A(1, 2) and C(5, 2) can be calculated as:
\[ \text{Length of AC} = |x_C - x_A| = |5 - 1| = 4 \text{ units} \]

- **Diagonal BD**:
- The distance between points B(3, 4) and D(3, 0) can be calculated as:
\[ \text{Length of BD} = |y_B - y_D| = |4 - 0| = 4 \text{ units} \]

### Conclusion
Since both diagonals AC and BD measure 4 units, we conclude that the diagonals of rhombus ABCD are equal.