Problem Analysis:
The given image shows a parallelogram \(ABCD\) with diagonals \(AC\) and \(BD\) intersecting at point \(O\). The lengths of segments \(AO\) and \(BO\) are given as 4 cm and 3 cm, respectively. We are tasked with solving a problem, which typically involves finding the area of the parallelogram or some related geometric property.
Key Properties to Use:
1.
Diagonals of a Parallelogram: The diagonals of a parallelogram bisect each other. This means:
- \(AO = OC = 4 \, \text{cm}\)
- \(BO = OD = 3 \, \text{cm}\)
2.
Area of a Parallelogram Using Diagonals: The area of a parallelogram can be calculated using the formula:
\[
\text{Area} = \frac{1}{2} \times d_1 \times d_2 \times \sin(\theta)
\]
where \(d_1\) and \(d_2\) are the lengths of the diagonals, and \(\theta\) is the angle between them. However, if the diagonals are perpendicular (i.e., \(\theta = 90^\circ\)), the formula simplifies to:
\[
\text{Area} = \frac{1}{2} \times d_1 \times d_2
\]
3.
Right Triangle Formation: Since the diagonals bisect each other, the intersection point \(O\) divides each diagonal into two equal parts. If we consider \(\triangle AOB\), it is a right triangle because the diagonals of a rhombus (a special case of a parallelogram) are perpendicular. We can use the Pythagorean theorem to find the side lengths of the parallelogram.
Step-by-Step Solution:
#### Step 1: Determine the lengths of the diagonals.
- Since \(AO = OC = 4 \, \text{cm}\), the full length of diagonal \(AC\) is:
\[
AC = AO + OC = 4 + 4 = 8 \, \text{cm}
\]
- Since \(BO = OD = 3 \, \text{cm}\), the full length of diagonal \(BD\) is:
\[
BD = BO + OD = 3 + 3 = 6 \, \text{cm}
\]
#### Step 2: Check if the diagonals are perpendicular.
- In \(\triangle AOB\), the sides are \(AO = 4 \, \text{cm}\), \(BO = 3 \, \text{cm}\), and \(AB\) is the hypotenuse. Using the Pythagorean theorem:
\[
AB^2 = AO^2 + BO^2
\]
\[
AB^2 = 4^2 + 3^2 = 16 + 9 = 25
\]
\[
AB = \sqrt{25} = 5 \, \text{cm}
\]
- Since \(AB\) is the hypotenuse, \(\triangle AOB\) is a right triangle, and the diagonals \(AC\) and \(BD\) are perpendicular.
#### Step 3: Calculate the area of the parallelogram.
- Since the diagonals are perpendicular, the area of the parallelogram is given by:
\[
\text{Area} = \frac{1}{2} \times d_1 \times d_2
\]
where \(d_1 = AC = 8 \, \text{cm}\) and \(d_2 = BD = 6 \, \text{cm}\):
\[
\text{Area} = \frac{1}{2} \times 8 \times 6 = \frac{1}{2} \times 48 = 24 \, \text{cm}^2
\]
Final Answer:
\[
\boxed{24}
\]
Parent Tip: Review the logic above to help your child master the concept of area of rhombus worksheet.
