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 $$ m \overline{\mathrm{QR}}=12 \mathrm{~cm} $$

Question

UpStudy AI · Accepted Answer

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!
Triangle PQR is a right-angled triangle with the right angle at Q. The sides are PQ = 16 cm, QR = 12 cm, and PR = 20 cm. The area is 96 cm², the perimeter is 48 cm, and the angles are approximately 36.87° at P and 53.13° at R.

PQR is a right angled triangle inwhich ,
and

Solución de inteligencia artificial de Upstudy

Responder

Solución

Bonus Knowledge

preguntas relacionadas

Latest Geometry Questions

PQR is a right angled triangle inwhich , and