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Calculate Additional Wood Needed for an Isosceles Right Triangle Art Project (92 cm Hypotenuse)
Mathematics
Grade 9 (Junior High School)
Question Content
11) Lorena and Karla are creating an art project in the shape of a right triangle. They have a 92 cm long piece of wood, which is to be used for the hypotenuse. The two legs of the triangular support are of equal length. Approximately how many more centimeters of wood do they need to complete the support? Give answer approximately.
Correct Answer
Approximately 130.1 centimeters
Detailed Solution Steps
1
Step 1: Define variables. Let x be the length of each equal leg of the right isosceles triangle, and the hypotenuse is given as 92 cm.
2
Step 2: Use the Pythagorean theorem for the right isosceles triangle: x² + x² = 92². Simplify the left side to 2x² = 8464.
3
Step 3: Solve for x. Divide both sides by 2: x² = 4232. Take the square root: x = √4232 = √(16×264.5) = 4√264.5 ≈ 65.05 cm.
4
Step 4: Calculate total wood needed for the support. Add the lengths of the two legs: 2x ≈ 2×65.05 = 130.1 cm. This is the total additional wood required.
Knowledge Points Involved
1
Right Isosceles Triangle Properties
A right isosceles triangle has one right angle and two equal-length legs. The hypotenuse is √2 times the length of either leg, derived directly from the Pythagorean theorem.
2
Pythagorean Theorem for Isosceles Right Triangles
For a right isosceles triangle with leg length x, the theorem simplifies to 2x² = c², where c is the hypotenuse. This specialized formula speeds up calculations for triangles with equal legs.
3
Approximate Square Root Calculation
When a square root does not result in an integer, use a calculator or prime factorization to find a decimal approximation, which is used for real-world measurement problems requiring approximate values.
4
Real-World Geometric Problem Solving
Applying geometric theorems and formulas to practical scenarios, such as calculating materials needed for a construction or art project, by translating real objects into geometric shapes and solving for unknown measurements.
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