Identify the geometric shapes and provide the missing values. Image description: A right-angled triangle ABC is shown. Angle S is marked as a right angle. The side RS has a length of 4.2 cm. The side PR has a length of 8.4 cm. The question asks to find the angles P and R.

This is a geometry problem involving a right-angled triangle.

Analysis:

• We are given a right-angled triangle with the right angle at S.
• The length of the side RS is 4.2 cm.
• The length of the hypotenuse PR is 8.4 cm.
• We need to find the angles P and R.

Calculations:

In a right-angled triangle, we can use trigonometric ratios.

For angle P:

• The side opposite to angle P is RS (length = 4.2 cm).
• The hypotenuse is PR (length = 8.4 cm).
• We can use the sine function: \[ \sin(P) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{RS}{PR} \]
• \[ \sin(P) = \frac{4.2}{8.4} = 0.5 \]
• To find the angle P, we take the inverse sine (arcsin):
• \[ P = \arcsin(0.5) \]
• \[ P = 30^{\circ} \]

For angle R:

• The sum of angles in a triangle is 180°.
• In a right-angled triangle, the sum of the two acute angles is 90°.
• \[ P + R + S = 180^{\circ} \]
• \[ 30^{\circ} + R + 90^{\circ} = 180^{\circ} \]
• \[ R + 120^{\circ} = 180^{\circ} \]
• \[ R = 180^{\circ} - 120^{\circ} \]
• \[ R = 60^{\circ} \]

Alternatively, for angle R:

• The side adjacent to angle R is RS (length = 4.2 cm).
• The hypotenuse is PR (length = 8.4 cm).
• We can use the cosine function: \[ \cos(R) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{RS}{PR} \]
• \[ \cos(R) = \frac{4.2}{8.4} = 0.5 \]
• To find the angle R, we take the inverse cosine (arccos):
• \[ R = \arccos(0.5) \]
• \[ R = 60^{\circ} \]

Ответ:

<p = 30°

<R = 60°