The area of a rhombus is 143 sqm. If the longer diagonal is 26 m, find the angle opposite the shorter diagonal and the length of one side of the rhombus

To solve this problem, let's first recall the formula for the area of a rhombus. The area \( A \) of a rhombus can be calculated using the lengths of its diagonals \( d_1 \) and \( d_2 \):

\[
A = \frac{1}{2} \times d_1 \times d_2
\]

Where:
- \( A = 143 \, \text{sqm} \) (the given area),
- \( d_1 = 26 \, \text{m} \) (the longer diagonal),
- \( d_2 \) is the unknown shorter diagonal.

### Step 1: Calculate the shorter diagonal \( d_2 \)

Substitute the known values into the formula and solve for \( d_2 \):

\[
143 = \frac{1}{2} \times 26 \times d_2
\]

\[
143 = 13 \times d_2
\]

\[
d_2 = \frac{143}{13} = 11 \, \text{m}
\]

### Step 2: Calculate the side length of the rhombus

The diagonals of a rhombus bisect each other at right angles, so we can form two right triangles. The sides of these triangles are half of each diagonal and the side length \( s \) of the rhombus is the hypotenuse of the triangle.

Half of the diagonals are:
\[
\frac{d_1}{2} = \frac{26}{2} = 13 \, \text{m} \quad \text{and} \quad \frac{d_2}{2} = \frac{11}{2} = 5.5 \, \text{m}
\]

Now, using the Pythagorean theorem:
\[
s = \sqrt{\left( \frac{d_1}{2} \right)^2 + \left( \frac{d_2}{2} \right)^2}
\]
\[
s = \sqrt{13^2 + 5.5^2}
\]
\[
s = \sqrt{169 + 30.25} = \sqrt{199.25} \approx 14.12 \, \text{m}
\]

So, the length of one side of the rhombus is approximately \( 14.12 \, \text{m} \).

### Step 3: Find the angle opposite the shorter diagonal

In the right triangle formed by half of each diagonal, we can use trigonometry to find the angle \( \theta \) opposite the shorter diagonal. Specifically, we use the tangent function:

\[
\tan(\theta) = \frac{\frac{d_2}{2}}{\frac{d_1}{2}} = \frac{5.5}{13}
\]
\[
\theta = \tan^{-1} \left( \frac{5.5}{13} \right)
\]

Let's calculate this:

\[
\theta \approx \tan^{-1} (0.4231) \approx 22.94^\circ
\]

Thus, the angle opposite the shorter diagonal is approximately \( 22.94^\circ \).

### Final answers:
- The length of one side of the rhombus is approximately \( 14.12 \, \text{m} \).
- The angle opposite the shorter diagonal is approximately \( 22.94^\circ \).

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Would you like further details or have any questions? 

Here are some related questions you might find useful:
1. How do you calculate the area of a rhombus using its side and angle?
2. What is the relationship between the diagonals of a rhombus?
3. How do you find the perimeter of a rhombus given the diagonals?
4. Can a rhombus have equal diagonals?
5. What other properties do the angles in a rhombus have?

**Tip:** For rhombuses, diagonals always bisect each other at right angles, which is a useful fact for solving many problems!

The area of a rhombus is 143 sqm. If the longer diagonal is 26 m, find the angle opposite the shorter diagonal and the length of one side of the rhombus.

Find the side length and angle of a rhombus with an area of 143 sqm and a diagonal of 26 m. This step-by-step guide covers geometry, the Pythagorean theorem, and trigonometric functions. Ideal for students in Grades 9-11.

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