Certainly! Let's go through the steps to solve each part of the problem.
### Part (a)
Given:
- Rhombus ABCD
- Diagonal AC = 16 cm
- Diagonal BD = 8 cm
We know that in a rhombus, the diagonals bisect each other at right angles. This means that the midpoint of both diagonals will be the same.
Let's consider the midpoint of the rhombus as the origin (0, 0).
#### Step-by-Step Solution:
1. Find coordinates of A and C:
- Diagonal AC is 16 cm long. This means that each half of the diagonal will be 8 cm.
- Since the midpoint is at (0, 0), point A will be 8 cm to the right of the origin, and point C will be 8 cm to the left of the origin.
Coordinates:
- A = (8, 0)
- C = (-8, 0)
2. Find coordinates of B and D:
- Diagonal BD is 8 cm long. This means that each half of the diagonal will be 4 cm.
- Since the midpoint is at (0, 0), point B will be 4 cm above the origin, and point D will be 4 cm below the origin.
Coordinates:
- B = (0, 4)
- D = (0, -4)
So, the coordinates of the vertices of the rhombus are:
- A = (8, 0)
- B = (0, 4)
- C = (-8, 0)
- D = (0, -4)
### Part (b)
Given coordinates:
- A (1, 3)
- B (1, -1)
- C (7, -1)
- D (7, 3)
We need to find the coordinates of the point of intersection of the diagonals.
#### Step-by-Step Solution:
1. Find the midpoint of diagonal AC:
- The coordinates of A are (1, 3)
- The coordinates of C are (7, -1)
To find the midpoint, we use the midpoint formula:
[tex]\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
Substituting the coordinates of A and C:
[tex]\[
\text{Midpoint of AC} = \left( \frac{1 + 7}{2}, \frac{3 + (-1)}{2} \right) = (4.0, 1.0)
\][/tex]
2. Find the midpoint of diagonal BD:
- The coordinates of B are (1, -1)
- The coordinates of D are (7, 3)
Using the same midpoint formula:
[tex]\[
\text{Midpoint of BD} = \left( \frac{1 + 7}{2}, \frac{-1 + 3}{2} \right) = (4.0, 1.0)
\][/tex]
So, the coordinates of the point of intersection of the diagonals are:
- (4.0, 1.0)
These coordinates are consistent across both pairs of diagonals, confirming the correctness of the solution.
In summary:
- The coordinates of A, B, C, D for the rhombus with diagonals AC = 16 cm and BD = 8 cm are:
- A = (8, 0)
- B = (0, 4)
- C = (-8, 0)
- D = (0, -4)
- The coordinates of the point of intersection of the diagonals for A(1, 3), B(1, -1), C(7, -1), D(7, 3) are:
- (4.0, 1.0)
