PQR is a right angled triangle inwhich \( m \angle \mathrm{Q}=90^{\circ} \), \( m \overline{\mathrm{PR}}=20 \mathrm{~cm}, m \overline{\mathrm{PQ}}=16 \mathrm{~cm} \) and \( m \overline{\mathrm{QR}}=12 \mathrm{~cm} \)

Yes, let's analyze **Triangle PQR**, which is a right-angled triangle with the right angle at vertex **Q**. ### Given: - **PR** (hypotenuse) = 20 cm - **PQ** = 16 cm - **QR** = 12 cm ### Verification of Right Angle Using the Pythagorean Theorem: For a right-angled triangle, the square of the hypotenuse should equal the sum of the squares of the other two sides. \[ PQ^2 + QR^2 = PR^2 \\ 16^2 + 12^2 = 20^2 \\ 256 + 144 = 400 \\ 400 = 400 \quad \checkmark \] This confirms that Triangle PQR is indeed a right-angled triangle with the right angle at **Q**. ### Calculating Other Properties: 1. **Area of Triangle PQR:** The area \( A \) of a right-angled triangle can be calculated using the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] Taking **PQ** as the base and **QR** as the height: \[ A = \frac{1}{2} \times 16 \, \text{cm} \times 12 \, \text{cm} = 96 \, \text{cm}^2 \] 2. **Perimeter of Triangle PQR:** The perimeter \( P \) is the sum of all its sides: \[ P = PQ + QR + PR = 16 \, \text{cm} + 12 \, \text{cm} + 20 \, \text{cm} = 48 \, \text{cm} \] 3. **Calculating the Other Angles:** - **Angle at P (\( \angle P \)):** Using the tangent function: \[ \tan(\angle P) = \frac{\text{Opposite side}}{\text{Adjacent side}} = \frac{QR}{PQ} = \frac{12}{16} = \frac{3}{4} \] \[ \angle P = \arctan\left(\frac{3}{4}\right) \approx 36.87^\circ \] - **Angle at R (\( \angle R \)):** Since the sum of angles in a triangle is \( 180^\circ \), and one angle is \( 90^\circ \): \[ \angle R = 90^\circ - \angle P \approx 90^\circ - 36.87^\circ = 53.13^\circ \] ### Summary: - **Type of Triangle:** Right-angled triangle at Q. - **Area:** 96 cm² - **Perimeter:** 48 cm - **Angles:** - \( \angle Q = 90^\circ \) - \( \angle P \approx 36.87^\circ \) - \( \angle R \approx 53.13^\circ \) If you have any specific questions or need further calculations related to Triangle PQR, feel free to ask!