Diagonals Answer Type Questions-1 the angles of the rhombus.

Certainly! Below are **Answer Type Questions** related to the **diagonals and angles of a rhombus**, along with their detailed solutions. These questions are designed to help you understand the properties of a rhombus and how its diagonals relate to its angles. --- ### **Question 1:** In a rhombus \( ABCD \), the lengths of the diagonals are \( d_1 = 24 \) cm and \( d_2 = 10 \) cm. Calculate all the interior angles of the rhombus. **Solution:** 1. **Properties of a Rhombus:** - All sides are equal. - Diagonals bisect each other at right angles. - Diagonals bisect the interior angles. 2. **Calculating Half-Angles:** - Since diagonals bisect the angles, consider one of the four right triangles formed by the diagonals. - The half-diagonals are: \[ \frac{d_1}{2} = \frac{24}{2} = 12 \text{ cm}, \quad \frac{d_2}{2} = \frac{10}{2} = 5 \text{ cm} \] - Let \( \theta \) be half of one of the acute angles. 3. **Using Trigonometry:** - In the right triangle, \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{5}{12} \) - Calculate \( \theta \): \[ \theta = \tan^{-1}\left(\frac{5}{12}\right) \approx 22.62^\circ \] - Therefore, the acute angle \( \alpha \): \[ \alpha = 2\theta \approx 2 \times 22.62^\circ = 45.24^\circ \] - Since the sum of adjacent angles in a rhombus is \( 180^\circ \), the obtuse angle \( \beta \): \[ \beta = 180^\circ - 45.24^\circ \approx 134.76^\circ \] 4. **Conclusion:** - All interior angles of the rhombus are: \[ 45.24^\circ \quad \text{and} \quad 134.76^\circ \] --- ### **Question 2:** If the diagonals of a rhombus are perpendicular and have lengths \( 14 \) cm and \( 48 \) cm, find the measure of each interior angle of the rhombus. **Solution:** 1. **Given:** - Diagonals \( d_1 = 14 \) cm and \( d_2 = 48 \) cm. 2. **Properties:** - Diagonals bisect each other at right angles. - Each diagonal splits the rhombus into four right-angled triangles. 3. **Calculating Half-Diagonals:** \[ \frac{d_1}{2} = 7 \text{ cm}, \quad \frac{d_2}{2} = 24 \text{ cm} \] 4. **Calculating Half-Angles:** - Let \( \theta \) be half of one of the acute angles. - Using trigonometry: \[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{7}{24} \] \[ \theta = \tan^{-1}\left(\frac{7}{24}\right) \approx 16.26^\circ \] - Therefore, the acute angle \( \alpha \): \[ \alpha = 2\theta \approx 2 \times 16.26^\circ = 32.52^\circ \] - The obtuse angle \( \beta \): \[ \beta = 180^\circ - 32.52^\circ = 147.48^\circ \] 5. **Conclusion:** - The interior angles of the rhombus are approximately: \[ 32.52^\circ \quad \text{and} \quad 147.48^\circ \] --- ### **Question 3:** A rhombus has one interior angle measuring \( 60^\circ \). Determine the lengths of its diagonals if each side of the rhombus is \( 10 \) cm. **Solution:** 1. **Given:** - Side length \( s = 10 \) cm. - One interior angle \( \alpha = 60^\circ \). 2. **Properties:** - The diagonals bisect the interior angles. - Let \( d_1 \) and \( d_2 \) be the diagonals. - Using the formula for the diagonals of a rhombus: \[ d_1 = 2s \sin\left(\frac{\alpha}{2}\right), \quad d_2 = 2s \cos\left(\frac{\alpha}{2}\right) \] 3. **Calculating Diagonals:** - Calculate \( \frac{\alpha}{2} = 30^\circ \). - \( d_1 = 2 \times 10 \times \sin(30^\circ) = 20 \times 0.5 = 10 \) cm. - \( d_2 = 2 \times 10 \times \cos(30^\circ) = 20 \times \left(\frac{\sqrt{3}}{2}\right) = 10\sqrt{3} \) cm. 4. **Conclusion:** - The lengths of the diagonals are: \[ d_1 = 10 \text{ cm}, \quad d_2 = 10\sqrt{3} \text{ cm} \approx 17.32 \text{ cm} \] --- ### **Question 4:** In rhombus \( WXYZ \), the diagonals intersect at point \( O \). If \( WX = 13 \) cm, \( WO = 5 \) cm, and \( XO = 12 \) cm, determine all the interior angles of the rhombus. **Solution:** 1. **Given:** - Side \( WX = 13 \) cm. - \( WO = 5 \) cm and \( XO = 12 \) cm. 2. **Understanding the Diagram:** - Diagonals \( WY \) and \( XZ \) intersect at \( O \). - \( WO = 5 \) cm implies \( WY = 10 \) cm (since diagonals bisect each other). - \( XO = 12 \) cm implies \( XZ = 24 \) cm. 3. **Determine Half-Angles:** - Consider triangle \( WOX \) which is right-angled (as diagonals of a rhombus are perpendicular). - Using Pythagoras theorem: \[ WX^2 = WO^2 + XO^2 \\ 13^2 = 5^2 + 12^2 \\ 169 = 25 + 144 \\ 169 = 169 \quad (\text{Verified}) \] - Calculate angle \( \theta \) at \( O \): \[ \theta = \angle XOW = \tan^{-1}\left(\frac{XO}{WO}\right) = \tan^{-1}\left(\frac{12}{5}\right) \approx 67.38^\circ \] - Therefore, the acute angle \( \alpha \): \[ \alpha = 2\theta \approx 2 \times 67.38^\circ = 134.76^\circ \] - The obtuse angle \( \beta \): \[ \beta = 180^\circ - 134.76^\circ = 45.24^\circ \] 4. **Conclusion:** - The interior angles of the rhombus are: \[ 134.76^\circ \quad \text{and} \quad 45.24^\circ \] --- ### **Key Takeaways:** - **Diagonals of a Rhombus:** - Bisect each other at right angles. - Bisect the interior angles. - **Calculating Angles:** - Use trigonometric ratios (sine, cosine, tangent) in the right-angled triangles formed by the diagonals to find the measures of the interior angles. - **Relationships:** - The acute and obtuse angles in a rhombus are supplementary (\( \alpha + \beta = 180^\circ \)). - Knowing the lengths of the diagonals allows for the calculation of the interior angles, and vice versa. I hope these questions and solutions help clarify how the diagonals and angles of a rhombus are related. If you have any further questions or need additional examples, feel free to ask!