Spatial Reasoning

Note: that the third trimester was the shortest of all of the trimesters, and mainly focused on chapter 8 (last post) and chapter 10 (current post). Everything else done in class after this point will be focused on prepping for the final.

OTHER KEY THINGS TO NOTE INCLUDE (DERIVING FROM ALL OF CHAPTER 10)

Three-dimensional figures, or solids, can be made up of flat or curved surfaces. Each flat surface is called a face. An edge is the segment that is the intersection of two faces. A vertex is the point that is the intersection of three or more faces.

A net is a diagram of the surfaces of a three-dimensional figure that can be folded to form the three-dimensional figure. To identify a three-dimensional figure from a net, look at the number of faces and the shape of each face.

In a perspective drawing, nonvertical parallel lines are drawn so that they meet at a point called a vanishing point. Vanishing points are located on a horizontal line called the horizon. A one-point perspective drawing contains one vanishing point. A two-point perspective drawing contains two vanishing points.

Prisms and cylinders have 2 congruent parallel bases.
A lateral face is not a base. The edges of the base are called base edges. A lateral edge is not an edge of a base. The lateral faces of a right prism are all rectangles. An oblique prism has at least one nonrectangular lateral face.

An altitude of a prism or cylinder is a perpendicular segment joining the planes of the bases. The height of a three-dimensional figure is the length of an altitude.

Surface area is the total area of all faces and curved
surfaces of a three-dimensional figure. The lateral
area of a prism is the sum of the areas of the lateral faces.

The vertex of a pyramid is the point opposite the base of the pyramid. The base of a regular pyramid is a regular polygon, and the lateral faces are congruent isosceles triangles. The slant height of a regular pyramid is the distance from the vertex to the midpoint of an edge of the base. The altitude of a pyramid is the perpendicular segment from the vertex to the plane of the base.

The vertex of a cone is the point opposite the base. The axis of a cone is the segment with endpoints at the vertex and the center of the base. The axis of a right cone is perpendicular to the base. The axis of an oblique cone is not perpendicular to the base.

The slant height of a right cone is the distance from the vertex of a right cone to a point on the edge of the base. The altitude of a cone is a perpendicular segment from the vertex of the cone to the plane of the base.

The volume of a three-dimensional figure is the number of nonoverlapping unit cubes of a given size that will exactly fill the interior.

Cavalieri’s principle says that if two three-dimensional figures have the same height and have the same cross-sectional area at every level, they have the same volume.

A sphere is the locus of points in space that are a fixed distance from a given point called the center of a sphere. A radius of a sphere connects the center of the sphere to any point on the sphere. A hemisphere is half of a sphere. A great circle divides a sphere into two hemispheres

Here are some visuals that help to describe each of the concepts above:



























Holt Geometry Book

Right Triangles and Trigonometry

Some visuals about right triangles and trigonometry:










More information:



Angles of elevation and depression:


An angle of elevation is the angle formed by a horizontal line and a line of sight to a point above the line. In the diagram, 1 is the angle of elevation from the tower T to the plane P.

An angle of depression is the angle formed by a horizontal line and a line of sight to a point below the line. 2 is the angle of depression from the plane to the tower.



Information about vectors:

Two vectors are equal vectors if they have the same
magnitude and the same direction

Two vectors are parallel vectors if they have the same direction or if they have opposite directions. They may have different magnitudes. For example, Equal vectors are always parallel vectors.

The resultant vector is the vector that represents the sum of two given vectors. To add two vectors geometrically, you can use the head-to-tail method or the parallelogram method.



Holt Geometry Book

The Amazing-ness of Circles

Some things to remember about circles and properties of circles:




Keep in mind: A central angle is an angle whose vertex is the center of a circle. An arc is an unbroken part of a circle consisting of two points called the endpoints and all the points on the circle between them.



Keep in mind that we just learned this chapter, and it is pretty fresh in my mind, so if I keep my comments to a bare minimum, you will know why. The circle information is pretty easy to understand, it's getting the hang of some of the problems that could be possibly difficult to some.





An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. An intercepted arc consists of endpoints that lie on the sides of an inscribed angle and all the points of the circle between them. A chord or arc subtends an angle if its endpoints lie on the sides of the angle.






Here is some information retained in chapter 11-5:





Entering into 11-6...


A secant segment is a segment of a secant with at least one endpoint on the circle. An external secant segment is a secant segment that lies in the exterior of the circle with one endpoint on the circle.



A tangent segment is a segment of a tangent with one endpoint on the circle. AB and AC are tangent segments.



The major formula from 11-7:


Holt Geometry Book