If the longer diagonal of a rhombus is 10 and the large angle is `100^(@)` , what is the area of the rhombus ?

Options

A. 37
B. 40
C. 42
D. 45

Explanation

To find the area of the rhombus given that the longer diagonal is 10 and the larger angle is \(100^\circ\), we can follow these steps:

### Step 1: Understand the properties of the rhombus
A rhombus has diagonals that bisect each other at right angles. This means that if we denote the diagonals as \(AC\) and \(BD\), they intersect at point \(O\) such that \(AO = OC\) and \(BO = OD\).

### Step 2: Identify the diagonals
Given that the longer diagonal \(BD = 10\), we can find the lengths of \(BO\) and \(OD\):
\[ BO = OD = \frac{BD}{2} = \frac{10}{2} = 5 \]

### Step 3: Analyze the angles
Since the diagonals bisect the angles of the rhombus, we can denote the larger angle \(A\) as \(100^\circ\). The angle \(BAO\) will be half of angle \(A\):
\[ \angle BAO = \frac{100^\circ}{2} = 50^\circ \]
Since the diagonals intersect at right angles, we have:
\[ \angle AOB = 90^\circ \]
Now, we can find angle \(ABO\) using the fact that the sum of angles in triangle \(ABO\) is \(180^\circ\):
\[ \angle ABO = 180^\circ - \angle BAO - \angle AOB = 180^\circ - 50^\circ - 90^\circ = 40^\circ \]

### Step 4: Use trigonometry to find \(AO\)
We can use the tangent function to find the length \(AO\):
\[ \tan(\angle ABO) = \frac{AO}{BO} \]
Substituting the known values:
\[ \tan(40^\circ) = \frac{AO}{5} \]
Thus,
\[ AO = 5 \tan(40^\circ) \]

### Step 5: Calculate \(AO\)
Using a calculator or trigonometric table, we find:
\[ \tan(40^\circ) \approx 0.8391 \]
So,
\[ AO \approx 5 \times 0.8391 \approx 4.1955 \]

### Step 6: Find the length of diagonal \(AC\)
Since \(AO\) is half of diagonal \(AC\):
\[ AC = 2 \times AO \approx 2 \times 4.1955 \approx 8.391 \]

### Step 7: Calculate the area of the rhombus
The area \(A\) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times AC \times BD \]
Substituting the values we found:
\[ A = \frac{1}{2} \times 8.391 \times 10 \approx \frac{83.91}{2} \approx 41.955 \]

### Final Answer
Thus, the area of the rhombus is approximately \(42\) square units.

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