A rectangle contains a strip of width $1,$ as shown below. Find the area of the strip.

[asy]
unitsize (1 cm);

pair A, B, C, D, E, F, P, Q;

A = (0,3);
B = (5,3);
C = (5,0);
D = (0,0);
E = (4,3);
F = (1,0);
P = interp(D,E,2/5);
Q = extension(P, P + rotate(90)*(D - E), B, F);

fill(B--E--D--F--cycle,gray(0.8));
draw(A--B--C--D--cycle);
draw(B--F);
draw(D--E);
draw((0,-0.2)--(5,-0.2),Arrows(6));
draw(P--Q,Arrows(6));

label("$7$", (A + D)/2, W, red);
label("$11$", (2.5,-0.2), S, red);
label("$1$", (P + Q)/2, NE, red);
[/asy]
	

To find the area of the strip of width $1$ in the rectangle, we first analyze the given dimensions and positioning of the strip within the rectangle. 

### Step-by-step Solution:

1. **Understand the Diagram and Geometry**:
   - The rectangle has a width of $7$ and a length of $11$.
   - A strip of width $1$ lies diagonally inside the rectangle, bounded by two parallel lines.

2. **Analyze the Diagonal Lines**:
   - The strip is defined by two parallel diagonal lines crossing the rectangle. The distance between these parallel lines is $1$.

3. **Area of Parallelogram Formed by Diagonal Strip**:
   - The strip forms a parallelogram inside the rectangle. The width of the strip (i.e., the perpendicular distance between the two lines) is $1$.
   - To calculate the area of this strip, we need to determine the length of the segment that each line cuts across the rectangle (base of the parallelogram).

4. **Calculate the Diagonal Length of the Rectangle**:
   - The diagonal length $d$ of the rectangle can be calculated using the Pythagorean theorem:
   \[
   d = \sqrt{7^2 + 11^2} = \sqrt{49 + 121} = \sqrt{170}.
   \]

5. **Length of Base of the Parallelogram**:
   - Each diagonal line of the strip extends from one side of the rectangle to the opposite side. As these lines are diagonals:
   - The length of the base of the parallelogram is the same as the diagonal of the rectangle, $ \sqrt{170} $.

6. **Calculate the Area of the Strip**:
   - The area $A$ of the parallelogram (strip) with base $ \sqrt{170} $ and height $1$ is given by:
   \[
   A = \text{base} \times \text{height} = \sqrt{170} \times 1 = \sqrt{170}.
   \]

Therefore, the area of the strip is:

\[
\boxed{\sqrt{170}}.
\]

Learn how to calculate the area of a strip within a rectangle using geometry principles. This problem involves understanding parallelograms, applying the Pythagorean theorem, and calculating diagonal lengths. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation