Geometry Chapter 7 Flashcards

Questions and Answers

The _________ of a region is the number of square units needed to cover it completely.

area

What is the formula for the area of a square?

A = s²

What does the Area Addition Postulate state?

If the interiors of two regions do not intersect, then the area of their union is the sum of their areas.

What is the Congruent Regions Postulate?

Congruent regions have the same area. Signup and view all the answers

What is the formula for the area of a rectangle?

A = bh Signup and view all the answers

What is the formula for the area of a parallelogram?

A = bh Signup and view all the answers

What is the formula for the area of a triangle?

A = 1/2 bh Signup and view all the answers

What is the formula for the area of a trapezoid?

A = 1/2 h(b₁ + b₂) Signup and view all the answers

What is the formula for the area of a kite?

A = 1/2 d₁d₂ Signup and view all the answers

What is the formula for the area of a rhombus?

A = 1/2 d₁d₂ Signup and view all the answers

What does the Pythagorean Theorem state in a right triangle?

a² + b² = c² Signup and view all the answers

What is the Converse of the Pythagorean Theorem?

If the sum of the squares of the lengths of two sides of a triangle equals the square of the length of the third side, then the triangle is a right triangle. Signup and view all the answers

What is the Corollary of the Pythagorean Theorem?

A triangle with side lengths a, b, and c; where a ≤ c and b ≤ c, is an acute triangle if c² < a² + b² and is an obtuse triangle if c² > a² + b². Signup and view all the answers

What does the 45-45-90 Triangle Theorem state?

If the length of a leg of an isosceles triangle is 'l', then the length of the hypotenuse is l√2. Signup and view all the answers

What does the 30-60-90 Triangle Theorem state?

If the length of the shorter leg of a 30-60-90 triangle is 'x', then the length of the longer leg is x√3 and the length of the hypotenuse is 2x. Signup and view all the answers

What is the formula for the area of an equilateral triangle?

A = √3/4 s² Signup and view all the answers

The _______ of a regular polygon is the segment that joins the center and a vertex of the polygon.

radius Signup and view all the answers

A _________ of a regular polygon is the angle formed by two radii drawn to consecutive vertices.

central angle Signup and view all the answers

An _________ of a regular polygon is the perpendicular segment that joins the center with a side of the polygon.

apothem Signup and view all the answers

The radii of a regular n-gon divide the polygon into n congruent isosceles triangles and form congruent central angles with measures of ________.

360⁰/n Signup and view all the answers

What is the formula for the area of a regular polygon?

A = 1/2 ap Signup and view all the answers

What is the formula for the area of a circle?

A = πr² Signup and view all the answers

A circle's __________ is the angle whose vertex is the center of the circle and whose sides contain radii of the circle.

central angle Signup and view all the answers

A _________ of a circle is the region bounded by two radii and the intercepted arc.

sector Signup and view all the answers

A ________ of a circle is the region bounded by a chord and its intercepted arc.

segment Signup and view all the answers

What is the formula for the area of a sector of a circle?

A = θ/360 πr² Signup and view all the answers

What is the formula for the apothem of equilateral triangles when you have the value of the height?

a = 1/3 h Signup and view all the answers

What is the formula for the apothem of equilateral triangles when you have the value of the side?

a = √3/6 s Signup and view all the answers

What is the formula for the radius of equilateral triangles when you have the value of the side?

r = √3/3 s Signup and view all the answers

Study Notes

Area Concepts

Area is defined as the total number of square units required to completely cover a region.
The area of a square can be calculated using the formula A = s², where s is the length of a side.
Area Addition Postulate states that if two regions do not overlap, the area of their union equals the sum of their areas.
Congruent Regions Postulate asserts that regions that are congruent share the same area.

Area Formulas for Common Shapes

For a rectangle, area is calculated using A = bh, where b is the base and h is the height.
A parallelogram also uses the formula A = bh for area calculation, with the same definitions for base and height.
The area of a triangle is determined with A = 1/2 bh.
A trapezoid has an area of A = 1/2 h(b₁ + b₂), where b₁ and b₂ are the lengths of the bases.
Kites and rhombuses can both be measured using the formula A = 1/2 d₁d₂, where d₁ and d₂ are the lengths of the diagonals.

Theorems and Relationships

The Pythagorean Theorem states that in a right triangle, a² + b² = c², where c is the length of the hypotenuse.
The Converse of the Pythagorean Theorem establishes that if the sum of the squares of two sides is equal to the square of the third side, the triangle is right-angled.
A Corollary of the Pythagorean Theorem classifies triangles: acute if c² < a² + b² and obtuse if c² > a² + b².

Special Triangles

In a 45-45-90 triangle, if one leg is l, the hypotenuse measures l√2.
In a 30-60-90 triangle, if the shorter leg is x, the longer leg is x√3 and the hypotenuse is 2x.
The area of an equilateral triangle is given by A = √3/4 s², where s is the length of one side.

Properties of Polygons

The radius of a regular polygon connects the center to a vertex.
A central angle in a regular polygon is formed by two radii and measures 360⁰/n for a polygon with n sides.
An apothem refers to the perpendicular distance from the center to a side of the polygon.
The area of a regular polygon is calculated with A = 1/2 ap, where a is the apothem and p is the perimeter.

Circular Geometry

The area of a circle is computed using A = πr², where r is the radius.
A sector is defined as the area bounded by two radii and an arc of the circle.
A segment of a circle is the region defined by a chord and the arc it intercepts.
The area of a sector can be calculated with the formula A = θ/360 πr², where θ is the central angle in degrees.

Calculating Apothem and Radius for Equilateral Triangles

To find the apothem of an equilateral triangle using height: a = 1/3 h.
For finding the apothem from the side length, use a = √3/6 s.
The radius from the side length in equilateral triangles is determined with r = √3/3 s.

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Test your knowledge of key concepts in Geometry Chapter 7 with these flashcards. This quiz covers important definitions, formulas, and postulates related to area. Perfect for anyone looking to reinforce their understanding of geometric principles.