Perimeter of the quadrilateral bounded by the coordinate axis and the lines `x+y=50` and `3x+y=90` is

Options

A. `80+20sqrt(2)`
B. `80+10sqrt(10)`
C. `80+20sqrt(2)+10sqrt(10)`
D. 110

Answer

To find the perimeter of the quadrilateral bounded by the coordinate axes and the lines \( x + y = 50 \) and \( 3x + y = 90 \), we will follow these steps:

### Step 1: Find the intercepts of the line \( x + y = 50 \)

1. **Find the x-intercept**: Set \( y = 0 \).
\[ x + 0 = 50 \implies x = 50 \]
So, the x-intercept is \( (50, 0) \).

2. **Find the y-intercept**: Set \( x = 0 \).
\[ 0 + y = 50 \implies y = 50 \]
So, the y-intercept is \( (0, 50) \).

### Step 2: Find the intercepts of the line \( 3x + y = 90 \)

1. **Find the x-intercept**: Set \( y = 0 \).
\[ 3x + 0 = 90 \implies 3x = 90 \implies x = 30 \]
So, the x-intercept is \( (30, 0) \).

2. **Find the y-intercept**: Set \( x = 0 \).
\[ 3(0) + y = 90 \implies y = 90 \]
So, the y-intercept is \( (0, 90) \).

### Step 3: Find the intersection of the lines \( x + y = 50 \) and \( 3x + y = 90 \)

To find the intersection point, we can solve the two equations simultaneously.

1. From \( x + y = 50 \), we can express \( y \) in terms of \( x \):
\[ y = 50 - x \]

2. Substitute \( y \) in the second equation:
\[ 3x + (50 - x) = 90 \]
Simplifying this gives:
\[ 3x + 50 - x = 90 \implies 2x = 40 \implies x = 20 \]

3. Substitute \( x = 20 \) back into \( y = 50 - x \):
\[ y = 50 - 20 = 30 \]
So, the intersection point is \( (20, 30) \).

### Step 4: Identify the vertices of the quadrilateral

The quadrilateral is formed by the points:
- \( A(50, 0) \)
- \( B(0, 50) \)
- \( C(0, 90) \)
- \( D(30, 0) \)
- \( E(20, 30) \) (intersection point)

### Step 5: Calculate the lengths of the sides of the quadrilateral

1. **Length \( AB \)**:
\[ AB = \sqrt{(50 - 0)^2 + (0 - 50)^2} = \sqrt{50^2 + 50^2} = \sqrt{2500 + 2500} = \sqrt{5000} = 50\sqrt{2} \]

2. **Length \( BC \)**:
\[ BC = 90 - 50 = 40 \]

3. **Length \( CD \)**:
\[ CD = \sqrt{(30 - 0)^2 + (0 - 90)^2} = \sqrt{30^2 + 90^2} = \sqrt{900 + 8100} = \sqrt{9000} = 30\sqrt{10} \]

4. **Length \( DA \)**:
\[ DA = \sqrt{(50 - 30)^2 + (0 - 0)^2} = \sqrt{20^2} = 20 \]

### Step 6: Calculate the perimeter

The perimeter \( P \) of the quadrilateral is the sum of the lengths of its sides:
\[ P = AB + BC + CD + DA \]
Substituting the values we found:
\[ P = 50\sqrt{2} + 40 + 30\sqrt{10} + 20 \]
\[ P = 50\sqrt{2} + 60 + 30\sqrt{10} \]

### Final Answer

Thus, the perimeter of the quadrilateral is \( 50\sqrt{2} + 60 + 30\sqrt{10} \).

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