What’s the answer ---**Extraction Content:**
**Question Stem:**
58. It takes 24 feet of fence to surround a square piece of land in Linda's backyard. Linda wants to plant flowers everywhere but on the diagonal sidewalk (shaded in the diagram below). Note that there is 1 foot of sidewalk along two sides of the square as indicated in the diagram below. What is the area of the sidewalk?
**Options:**
(A) 5 ft²
(B) 5.5 ft²
(C) 6 ft²
(D) 6.5 ft²
(E) 7 ft²
**Other Relevant Text:**
Problem number: 58
Page number: 18
**Chart/Diagram Description:**
* **Type:** Geometric diagram showing a square with a shaded diagonal parallelogram inside.
* **Main Elements:**
* A square shape is depicted.
* Outer dimensions of the square are indicated by braces and labels implicitly showing a side length. A label "6 Ft" is written above a brace spanning the top edge. A brace spans the right edge and is labeled "6 Ft" (implied by the square shape). The text confirms it's a square with a perimeter of 24 feet, so each side is 6 feet.
* Along the bottom edge, a brace from the left corner spans a length labeled "1 foot". Another brace spans the remaining length to the right corner, labeled "5 Ft". The total length 1 + 5 = 6 feet matches the side length.
* Along the left edge, a brace from the bottom corner spans a length labeled "1 Ft". Another brace spans the remaining length upwards, labeled "5 Ft". The total length 1 + 5 = 6 feet matches the side length.
* A region inside the square is shaded, representing a diagonal sidewalk.
* The shaded region is a parallelogram, running from the bottom-left corner area towards the top-right corner area.
* A brace is shown across the shaded diagonal region, labeled "1 foot", indicating its width measured perpendicular to the diagonal direction.
* The unshaded areas near the bottom-left and top-right corners are where flowers are planted. The diagram suggests the unshaded areas include the triangles formed by the sides and lines connecting the 1-foot marks on the adjacent sides (e.g., the triangle with vertices at the bottom-left corner, the 1-foot mark on the bottom side, and the 1-foot mark on the left side). However, the shaded region starts exactly at the bottom-left corner in the diagram.
**Mathematical Formulas/Equations:**
None explicitly present, but implied geometric area calculation. Perimeter of square = 4 * side length. Area of square = side length². Area of parallelogram = base * height.
**Problem Number:** 58
**Question Stem:**
It takes 24 feet of fence to surround a square piece of land in Linda's backyard. Linda wants to plant flowers everywhere but on the diagonal sidewalk (shaded in the diagram below). Note that there is 1 foot of sidewalk along two sides of the square as indicated in the diagram below. What is the area of the sidewalk?
**Diagram Description:**
* Type: Geometric figure, representing a square piece of land with a diagonal sidewalk.
* Main Elements:
* A square representing the land.
* A diagonal line segment drawn from one corner to the opposite corner.
* A shaded region along the diagonal line, representing the sidewalk.
* Annotations indicating measurements:
* A curved bracket along the left side of the square with the label "6 ft".
* A curved bracket along the bottom side of the square with the label "6 ft".
* A curly bracket indicated next to the corner where the sidewalk starts, along the top edge, labeled "1 foot".
* A curly bracket indicated next to the corner where the sidewalk starts, along the left edge, labeled "1 ft". (This label "1 ft" is next to the "6 ft" bracket on the left side and seems to indicate a segment length, likely the width of the sidewalk extension at the corner).
* A curly bracket along the right edge of the square, from the corner where the sidewalk ends upwards, labeled "5 Ft".
* A curved bracket along the bottom edge of the square, from the corner where the sidewalk ends towards the left, labeled "5 Ft".
* The shaded area representing the sidewalk is along the diagonal. Its width appears consistent. The diagram suggests the sidewalk extends 1 foot along two adjacent sides from the corner, forming a small triangle at the corner, and then continues diagonally. However, the question states "1 foot of sidewalk along two sides of the square as indicated". The diagram shows 1 foot length measured along the side from the corner where the diagonal starts. The diagonal sidewalk is shaded.
**Options:**
(A) 5 ft²
(B) 5.5 ft²
(C) 6 ft²
(D) 6.5 ft²
(E) 7 ft²
**Other Relevant Text:**
"18" (Number at the bottom left, likely page number)
视频信息
答案文本
视频字幕
Let's analyze this geometry problem step by step. We have a square piece of land that requires 24 feet of fence to surround it. Since the perimeter of a square is 4 times the side length, each side must be 6 feet long. The problem shows a diagonal sidewalk running across the square, with a width of 1 foot along two adjacent sides from the corner.
Now let's understand the shape of the sidewalk. The sidewalk consists of two parts: first, a small triangle at the bottom-left corner formed by the 1-foot extensions along the bottom and left edges. Second, a diagonal strip that runs from this corner triangle to the opposite corner, maintaining a constant width of 1 foot perpendicular to the diagonal direction.
Let's calculate the area step by step. First, the corner triangle has legs of 1 foot each, so its area is one-half times 1 times 1, which equals 0.5 square feet. Next, we need the diagonal length. Since the remaining sides are 5 feet each, the diagonal length is the square root of 5 squared plus 5 squared, which equals 5 times the square root of 2 feet. The diagonal strip area is this length times the width of 1 foot, giving us 5 square root 2 square feet.
Now let's complete the calculation. The total sidewalk area is 0.5 plus 5 square root 2. Since square root 2 is approximately 1.414, we get 0.5 plus 5 times 1.414, which equals 0.5 plus 7.07, giving us 7.57 square feet. Looking at the answer choices, this is closest to option E, which is 7 square feet. Therefore, the answer is E.
To summarize our solution: We started with a square having 24 feet perimeter, giving us 6-foot sides. The diagonal sidewalk consists of a corner triangle with area 0.5 square feet, plus a diagonal strip with area 5 square root 2 square feet. The total area is approximately 7.57 square feet, making option E the correct answer with 7 square feet.