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Find the Perimeter of a Triangular Grass Section in a Square Backyard (Area 225 sq ft)
Mathematics
Grade 8 (Junior High School)
Question Content
10) AMike has a square backyard with an area of 225 ft sq. He started by splitting his yard diagonally to plant grass seed but only did half his yard. What is the perimeter of the grass section of the backyard? Give answers in simplest radical form.
Correct Answer
(30 + 30√2) feet
Detailed Solution Steps
1
Step 1: Find the side length of the square backyard. Since the area of a square is side length squared (A = s²), set up the equation s² = 225. Taking the square root of both sides, s = √225 = 15 feet.
2
Step 2: Identify the grass section shape. Splitting the square diagonally creates two congruent right isosceles triangles, so the grass section is one of these triangles with two legs equal to the square's side length (15 ft) and a hypotenuse equal to the square's diagonal.
3
Step 3: Calculate the length of the square's diagonal. Using the Pythagorean theorem for the right triangle: diagonal (d) = √(15² + 15²) = √(225 + 225) = √450 = √(225×2) = 15√2 feet.
4
Step 4: Calculate the perimeter of the triangular grass section. Add the lengths of its three sides: 15 + 15 + 15√2 = 30 + 30√2 feet.
Knowledge Points Involved
1
Area of a Square
The area of a square is calculated by squaring the length of one of its sides, given by the formula A = s², where A is area and s is side length. This formula is used to reverse-engineer the side length when the area is known.
2
Pythagorean Theorem
For a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides, written as a² + b² = c², where c is the hypotenuse and a, b are the legs. It is used to find unknown side lengths in right triangles, including the diagonal of a square.
3
Simplest Radical Form
A radical expression is in simplest form when no perfect square factors other than 1 are under the radical sign, and there are no radicals in the denominator. For example, √450 simplifies to 15√2 by factoring out the perfect square 225.
4
Perimeter of a Triangle
The perimeter of a triangle is the total distance around the shape, calculated by adding the lengths of all three of its sides. For composite shapes formed from polygons, it uses the side lengths of the original polygon to find the perimeter of the smaller section.
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