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Questions and Answers
What does Postulate 8 or The Area Postulate state?
What does Postulate 8 or The Area Postulate state?
- Every polygonal region has a positive number called its area
- Congruent triangles have equal areas
- The area of a polygonal region is equal to the sum of the areas of its nonoverlapping parts
- All of the above (correct)
What is the formula for the area of a rectangle according to Postulate 9?
What is the formula for the area of a rectangle according to Postulate 9?
The area of a rectangle is the product of its base and altitude.
What is the area of a square according to the Corollary to Postulate 9?
What is the area of a square according to the Corollary to Postulate 9?
The area of a square is the square of its sides.
What does Theorem 38 state about the area of a right triangle?
What does Theorem 38 state about the area of a right triangle?
According to Theorem 39, how is the area of a triangle calculated?
According to Theorem 39, how is the area of a triangle calculated?
What does the Corollary to Theorem 39 imply about triangles with equal bases and altitudes?
What does the Corollary to Theorem 39 imply about triangles with equal bases and altitudes?
What is the area of a parallelogram according to Theorem 40?
What is the area of a parallelogram according to Theorem 40?
According to Theorem 41, how is the area of a trapezoid calculated?
According to Theorem 41, how is the area of a trapezoid calculated?
What does Theorem 42 or The Pythagorean Theorem state?
What does Theorem 42 or The Pythagorean Theorem state?
What is the Converse of the Pythagorean Theorem according to Theorem 43?
What is the Converse of the Pythagorean Theorem according to Theorem 43?
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Study Notes
Area Postulates and Theorems
- Postulate 8 (Area Postulate): Defines area as a positive number for every polygonal region, ensuring congruent triangles have equal areas and the total area is the sum of its non-overlapping parts.
- Postulate 9: States that the area of a rectangle can be calculated by multiplying its base by its altitude.
- Corollary to Postulate 9: Specifies that the area of a square is determined by squaring the length of its sides.
Triangle Area Theorems
- Theorem 38: Establishes that the area of a right triangle equals half the product of its two legs.
- Theorem 39: Generalizes that the area of any triangle can be calculated as half the product of any base and its corresponding altitude.
- Corollary to Theorem 39: Indicates that triangles with the same base and height will have equal areas.
Area of Parallelograms and Trapezoids
- Theorem 40: States that the area of a parallelogram is derived from the product of any chosen base and the corresponding altitude.
- Theorem 41: Describes the area of a trapezoid as half of the product of its altitude and the sum of its two bases.
Pythagorean Theorem
- Theorem 42 (Pythagorean Theorem): Relates the sides of a right triangle, asserting that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
- Theorem 43 (Converse of the Pythagorean Theorem): Determines that if one side of a triangle's square equals the sum of the squares of the other two sides, the triangle is classified as a right triangle.
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