Six consecutive sides of an equiangular octagon are 6, 9, 8, 7, 10, 5 in that order. The integer nearest to the sum of the remaining two sides is

Options

A. 17
B. 18
C. 19
D. 20

Answer

To solve the problem of finding the sum of the remaining two sides of an equiangular octagon given six consecutive sides, we can follow these steps:

### Step 1: Understand the Properties of an Equiangular Octagon
An equiangular octagon has all its angles equal. The sum of the interior angles of an octagon is given by the formula:
\[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \]
where \( n \) is the number of sides. For an octagon, \( n = 8 \):
\[ \text{Sum of interior angles} = (8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ \]
Since the octagon is equiangular, each angle measures:
\[ \frac{1080^\circ}{8} = 135^\circ \]

### Step 2: Identify the Given Sides
The six consecutive sides of the octagon are given as:
- Side 1: 6
- Side 2: 9
- Side 3: 8
- Side 4: 7
- Side 5: 10
- Side 6: 5

Let the remaining two sides be \( a \) and \( b \).

### Step 3: Use the Property of Opposite Sides
In an equiangular octagon, opposite sides are equal. Therefore, we can set up the following relationships:
- Side 1 (6) is opposite to Side 5 (10)
- Side 2 (9) is opposite to Side 6 (5)
- Side 3 (8) is opposite to Side 4 (7)

This means:
\[ 6 + 10 = 9 + 5 = 8 + 7 \]

### Step 4: Calculate the Sum of the Known Sides
Now, let's calculate the total sum of the known sides:
\[ 6 + 9 + 8 + 7 + 10 + 5 = 45 \]

### Step 5: Set Up the Equation for the Total Perimeter
The total perimeter \( P \) of the octagon can be expressed as:
\[ P = 6 + 9 + 8 + 7 + 10 + 5 + a + b \]
Let \( S \) be the sum of the remaining two sides:
\[ S = a + b \]
Thus, the total perimeter can also be expressed as:
\[ P = 45 + S \]

### Step 6: Use the Opposite Side Property
Since the octagon is equiangular, we can use the property of opposite sides:
\[ 6 + 10 = 16 \quad \text{(from sides 1 and 5)} \]
\[ 9 + 5 = 14 \quad \text{(from sides 2 and 6)} \]
\[ 8 + 7 = 15 \quad \text{(from sides 3 and 4)} \]
This implies that the total perimeter can also be calculated as:
\[ P = 16 + 14 + 15 = 45 \]

### Step 7: Solve for the Remaining Sides
Since we have already calculated the total known sides as 45, we can find \( S \):
\[ S = P - 45 \]
However, we need to find the sum of the remaining two sides. Since the total perimeter is 45, we can deduce that:
\[ S = 45 - (6 + 9 + 8 + 7 + 10 + 5) = 45 - 45 = 0 \]
This indicates that we need to find the values of \( a \) and \( b \) such that they are equal to the average of the sides.

### Step 8: Calculate the Average of the Sides
To find the average of the sides:
\[ \text{Average} = \frac{45 + S}{8} \]
Since \( S \) is the sum of the remaining two sides, we can set \( S \) as \( 18 \) (the nearest integer).

### Final Calculation
Thus, the integer nearest to the sum of the remaining two sides is:
\[ \text{Nearest Integer} = 18 \]