Use the diagram at the right. a. Find the area of □P Q R S . b. Find the area of P S R . c. Find

step 1 In this problem, we're first asked to find the area of the parallelogram, PQRS. The area of any

Use the diagram at the right. a. Find the area of □P Q R S ....

Use the diagram at the right. a. Find the area of $\square P Q R S$ . b. Find the area of $\triangle P S R$ . c. Find the area of $\triangle O S R$ . (Hint: Refer to $\triangle P S R$ and use Exercise 28 . $)$ d. What is the area of $\triangle P S O ?$ e. What must the area of $\triangle P O Q$ be? Why? What must the area of $\triangle O Q R$ be? f. State what you have shown in parts (a)-(e) about how the diagonals of a parallelogram divide the parallelogram.

Use the diagram at the right.
a. Find the area of $\square P Q R S$ .
b. Find the area of $\triangle P S R$ .
c. Find the area of $\triangle O S R$ . (Hint: Refer to $\triangle P S R$ and use Exercise 28 . $)$
d. What is the area of $\triangle P S O ?$
e. What must the area of $\triangle P O Q$ be? Why? What must the area of $\triangle O Q R$ be?
f. State what you have shown in parts (a)-(e) about how the diagonals of a parallelogram divide the parallelogram.

Key Concepts

Diagonals and Area Division

The diagonals in parallelograms not only bisect each other but also divide the parallelogram into triangles that are congruent or have equal areas. This insight is crucial in proving that multiple regions formed by the diagonals are identical in area, facilitating the solution of problems where one region's area can be inferred from another's.

Geometric Decomposition

Geometric decomposition refers to breaking a complex figure into simpler parts for which calculations are easier. By dissecting a parallelogram into triangles and other shapes using its diagonals, one can apply standard area formulas to each part and then relate these areas to understand the overall structure of the shape.

Area Calculation

This concept involves determining the area of various geometric figures such as squares and triangles. It uses formulas like side² for squares and ½ × base × height for triangles. Mastery of these formulas is essential for analyzing and decomposing more complex shapes into simpler ones whose areas can be computed.

Properties of Parallelograms

Parallelograms have special characteristics, such as opposite sides being parallel and equal in length, as well as diagonals that bisect each other. Understanding these properties helps in recognizing symmetry and equal area subdivisions within the shape, making the calculation of regions created by the diagonals straightforward.

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