Geometry

To determine the angle between the hour and minute hands when the time is "quarter past 3", follow these steps:

1. Minute hand position: When the time is 3:15, the minute hand is at the 15-minute mark. Each minute represents 6 degrees (since 360 degrees/60 minutes = 6 degrees per minute). Thus, the position is:
15 minutes × 6 degrees/minute = 90 degrees.

2. Hour hand position: The hour hand moves 30 degrees per hour (since 360 degrees/12 hours = 30 degrees per hour). At exactly 3:00, the hour hand is at 90 degrees (3 hours × 30 degrees/hour). Since 15 minutes is 1/4 of an hour, the hour hand moves a bit further:
(1/4) × 30 degrees = 7.5 degrees.
Thus, at 3:15, the hour hand is at:
90 degrees + 7.5 degrees = 97.5 degrees.

3. Calculating the angle between the hands: The angle between the hour and minute hands is the absolute difference between their positions:
|97.5 degrees - 90 degrees| = 7.5 degrees.

The angle made by the hour and minute hands at 3:15 is therefore degrees.

To solve the given problem, let's set up the equations based on the information:

We are given:

• ∠P + ∠R = 150° (Equation 1)
• ∠P + 3∠Q = 170° (Equation 2)

We know the sum of angles in a triangle is 180°:

∠P + ∠Q + ∠R = 180° (Equation 3)

To find ∠P, we will:

1. Express ∠R from Equation 1:
2. Substitute ∠R in Equation 3:
3. Now use the value of ∠Q in Equation 2:

Thus, ∠P is 80°.

To find the length of EC, we use the Basic Proportionality Theorem (also known as Thales' theorem), which states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.

Given that DE is parallel to BC, we apply the theorem to find:

We have:

Substitute the known values into the equation:

Cross-multiplying gives:

Solving for EC, we divide both sides by 6:

To find the area of the square, we first need to determine the side length of the square using the diagonals. Since the diagonals of a square are equal, we equate them:

Solving for :

1. Rearrange the equation:

2. Subtract from both sides:

3. Add to both sides:

4. Divide by :

Now substitute the value of back into one of the equations for the diagonal:

Length of diagonal = cm or cm

In a square, the relationship between the side length and the diagonal is given by:

So,

Solving for : cm

The area of the square is :

Area = cm²

The correct answer, therefore, is 162 cm².