Geometry Chapters 6-11 Flashcards

Questions and Answers

What is Chapter 6?

Geometry content.

Two polygons are similar if:

Corresponding sides are congruent (correct)

Corresponding angles are equal

Corresponding sides are equal

Corresponding angles are congruent (correct)

What is reflection in geometry?

Creates a mirror image by reflecting a figure over something.

What is rotation?

Turns a figure about a set point. Signup and view all the answers

What is translation?

Slides in any direction. Signup and view all the answers

Which are ways to prove triangles similar?

AA (Angle-Angle) Signup and view all the answers

What is the 45-45-90 theorem?

Triangle properties for equal legs and hypotenuse. Signup and view all the answers

What is the 30-60-90 theorem?

Hypotenuse = 2 * shorter leg; longer leg = shorter leg * √3. Signup and view all the answers

When do you use inverse trigonometry (sin-1)?

To find an angle measure. Signup and view all the answers

What are the interior angles of a quadrilateral?

360 degrees. Signup and view all the answers

How do you find the angle sum in a polygon?

(sides-2) * 180. Signup and view all the answers

What does the Polygon Exterior Angles Theorem state?

Exterior angles will add up to 360 degrees. Signup and view all the answers

What is a parallelogram?

A quadrilateral with both pairs of opposite sides parallel. Signup and view all the answers

What is a rhombus?

Parallelogram with four congruent sides and perpendicular diagonals. Signup and view all the answers

What is a rectangle?

Parallelogram with four right angles. Signup and view all the answers

What is a square?

Parallelogram with four right angles and four congruent sides. Signup and view all the answers

What is a trapezoid?

A quadrilateral with exactly one pair of parallel sides. Signup and view all the answers

When is a trapezoid considered isosceles?

If each pair of base angles is congruent and diagonals are congruent. Signup and view all the answers

What is the midsegment on a trapezoid?

1/2 length of bases combined. Signup and view all the answers

What are tangent circles?

Circles that touch at exactly one point. Signup and view all the answers

What are concentric circles?

Circles that have the same center but different radii. Signup and view all the answers

What is the standard equation of a circle?

(x-h)² + (y-k)² = r². Signup and view all the answers

How can you describe a translation?

(x,y) --> (x+a, y+b). Signup and view all the answers

What is the component form of a vector?

Horizontal then vertical. Signup and view all the answers

What is the reflection in the x-axis?

(x,y) --> (x, -y). Signup and view all the answers

What is the reflection in the y-axis?

(x,y) --> (-x, y). Signup and view all the answers

What is the result of reflecting a point in the line y=x?

(b,a). Signup and view all the answers

How is a point reflected in the line y=-x?

(-b, -a). Signup and view all the answers

What are the coordinates after rotating a point 90 degrees?

(-b, a). Signup and view all the answers

What happens when you rotate a point 180 degrees?

(-a, -b). Signup and view all the answers

What happens when you rotate a point 270 degrees?

(b, -a). Signup and view all the answers

What does the Reflections in Intersecting Lines Theorem state?

The angle of rotation is 2x. Signup and view all the answers

What is the circumference of a circle?

2πr or πd. Signup and view all the answers

What is the formula for arc length?

Length/2πr = Measure/360. Signup and view all the answers

What is the area of a circle?

πr². Signup and view all the answers

What is the area of a sector?

Area/πr² = Measure/360. Signup and view all the answers

How do you calculate the probability a point is on an area of a line?

Length of smaller/Length of larger. Signup and view all the answers

What is the area of a right cone?

πr² + πrl. Signup and view all the answers

What is the area of a cylinder?

2πr² + 2πrh. Signup and view all the answers

What is the area of a sphere?

4πr². Signup and view all the answers

What is the area of a trapezoid?

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

What is the volume of a cylinder?

πr²h. Signup and view all the answers

What is the volume of a solid?

Bh. Signup and view all the answers

What is the volume of a cone?

1/3πr²h. Signup and view all the answers

What is the volume of a pyramid?

1/3Bh. Signup and view all the answers

What is the volume of a sphere?

4/3πr³. Signup and view all the answers

What does Euler's Theorem state?

V + F = E + 2. Signup and view all the answers

What is a chord?

A segment whose endpoints are on the circle. Signup and view all the answers

What is a secant?

A line that intersects a circle in two points. Signup and view all the answers

What is a tangent?

Intersects the circle at exactly one point. Signup and view all the answers

What is a central angle?

An angle whose vertex is the center of the circle. Signup and view all the answers

What is the difference between minor arc and major arc?

Minor arc is shorter than the circle, major arc is longer. Signup and view all the answers

What are adjacent arcs?

Two arcs of the same circle that intersect at one point. Signup and view all the answers

What is an inscribed angle?

Angle whose vertex is on the circle and contains chords of the circle. Signup and view all the answers

Study Notes

Chapter 6: Basic Transformations and Similarity

Similar polygons have congruent corresponding angles and proportional corresponding sides.
Reflection creates a mirror image across a line or axis.
Rotation involves turning a figure around a specific point.
Translation refers to sliding a figure in any direction without altering its shape.

Chapter 7: Triangle Similarity

Triangles can be proven similar using:
- AA (Angle-Angle) postulate
- SSS (Side-Side-Side) similarity
- SAS (Side-Angle-Side) similarity.
The 45-45-90 theorem relates the lengths of the legs to the hypotenuse: hypotenuse = leg * √2.
In a 30-60-90 triangle, the relationships are:
- Hypotenuse = 2 × shorter leg
- Longer leg = shorter leg × √3.

Chapter 8: Angles in Polygons

The sum of interior angles in a quadrilateral is 360 degrees.
To find the angle sum of any polygon, use the formula: (sides - 2) * 180.
According to the Polygon Exterior Angles Theorem, the sum of exterior angles of any polygon equals 360 degrees.

Quadrilaterals Characteristics

A parallelogram has opposite sides that are equal and parallel, with bisecting diagonals and congruent opposite angles.
A rhombus is a type of parallelogram with four equal sides and perpendicular diagonals.
A rectangle is a parallelogram with four right angles.
A square is a special parallelogram that has both four right angles and four congruent sides.
A trapezoid features one pair of parallel sides known as bases.
An isosceles trapezoid has congruent base angles and equal diagonals.

Chapter 9 and 10: Circles and their Properties

Tangent circles touch at exactly one point.
Concentric circles share the same center but have different radii.
The standard equation of a circle with center (h, k) and radius r is given by: (x - h)² + (y - k)² = r².

Transformations and Vectors

A translation can be described by the notation: (x,y) → (x+a, y+b).
The component form of a vector consists of its horizontal and vertical movements.
Reflection in the x-axis transforms a point (x, y) to (x, -y).
Reflection in the y-axis transforms a point (x, y) to (-x, y).
Reflection in the line y = x changes a point (b, a) to (b, a).
Rotation transformations involve:
- 90 degrees: (-b, a)
- 180 degrees: (-a, -b)
- 270 degrees: (b, -a).

Chapter 11: Measurement of Circles and Solids

The circumference of a circle can be calculated using either 2πr or πd.
Arc length determination uses the ratio: Length/2πr = Measure/360.
The area of a circle can be found using the formula: πr².
The area of a sector of a circle follows: Area/πr² = Measure/360.

Areas and Volumes of Solids

The area of a right cone includes the base area (πr²) plus the lateral area (πrl).
The area of a cylinder combines the base areas and lateral surface area: 2πr² + 2πrh.
The surface area of a sphere is calculated as 4πr².
The area of a trapezoid is given by: 1/2h(b1+b2).
Volume calculations are characterized by:
- Cylinder: πr²h
- Cone: 1/3πr²h
- Pyramid: 1/3Bh
- Sphere: 4/3πr³.

Euler's Theorem and Circle Terminology

Euler's Theorem states the relation: V + F = E + 2, where V is vertices, F is faces, and E is edges.
A chord is a segment with endpoints on the circle.
A secant is a line that intersects a circle at two points.
A tangent touches a circle at exactly one point.
A central angle has its vertex at the circle's center.
Minor and major arcs are segments of a circle, defined by their angles.
Adjacent arcs are arcs that intersect at a single point.
An inscribed angle is formed by two chords from the circle.

Coordinate Geometry

The distance formula calculates the distance between two points in the coordinate plane.
The midpoint formula determines the midpoint between two endpoints on a line segment.
The slope is defined as the ratio of the rise over the run between two points on a line.

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Test your knowledge on key concepts from Geometry Chapters 6-11 with these flashcards. Each card focuses on important definitions and principles, including similarity of polygons, geometric transformations like reflection, rotation, and translation. Perfect for quick reviews or study sessions!