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 $22.94^\circ$.

---

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!