Geometry and measurement review

Geometry and Measurement Review

The SAT doesn’t include:

• Formal geometric proofs

• Trigonometry

• Radian measure

know: geometric

notation for points and lines, line segments, rays, angles and their measures, and lengths

Geometric Notation

Angles in the Plane

Vertical angles • two opposite angles formed by two

intersecting lines• have equal measureSupplementary angles• two angles whose sum is 180 degreesComplementary angles• two angles whose sum is 90 degrees

Triangles

Equilateral triangle• all three sides are equal length• all three angles measure 60 degrees

Isosceles triangle• two sides are equal in length• angles opposite the equal sides are equal

Right triangle• one right angle• hypotenuse is side opposite right

angle• hypotenuse is longest side• other two sides are called legs• leg2+leg2 = hypotenuse2 (Pythagorean

Theorem)

Special Right Triangles30°-60°-90° triangle

• short leg = x

• long leg = • hypotenuse = 2x

45°-45°-90° triangle

• legs are equal• angles opposite the legs are equal• each leg = x• hypotenuse =

3x

2x

Congruent triangles• all three pairs of corresponding

sides are congruent

• all three pairs of corresponding angles are congruent

• SSS, SAS, AAS, ASA

Similar Triangles

• same shape• lengths of corresponding sides

are in proportion • all pairs of corresponding angles

are congruent• AA

Triangle Inequality• sum of the lengths of any two sides of a

triangle is greater than the length of the third side

• When one side is length a and second side is length b, length of third side is between la –b l and a +b

• Ex: given sides of a , 10 and 16, third side is greater than 6 and less than 26

Quadrilaterals

Parallelograms• Opposite sides are congruent

• Opposite angles are congruent

• Consecutive angles are supplementary

Rectangles• parallelogram• all angles are right angles• diagonals are congruent

Squares

• rectangle and thus also parallelogram

• all sides are congruent

• diagonal is times the length of a side2

Areas and PerimetersRectangle• Area = l w• Perimeter = 2l + 2w Square• Area = s2

• Perimeter = 4sParallogram• Area = b h• Perimeter = 2l + 2w

Triangle

•

• Perimeter = sum of the three sides

Polygon

• Perimeter = sum of all the sidesRegular Polygon• all sides are equal length• all angles are equal measure

1Area

2b h

Angles in a Polygon Sum of interior angles: Triangle 180 Quadrilateral 360 Pentagon 540 Hexagon 720 n sides (n-2) 180

CirclesDiameter• line segment that passes through the

center and has its endpoints on the circle • all diameters in same circle are equal

lengthRadius• line segment from the center of the circle

to a point on the circle • all radii in same circle are equal length• or

1

2r d2r d

Central angle• angle whose vertex is the center of a

circle and formed by two radiiArc• part of a circle• measure is same as measure of

central angle that cuts the arc

Tangent to a circle

• a line that intersects the circle at exactly one point

• perpendicular to the radius at the point of tangency

Circumference of Circle

• distance around a circle

•

Area of Circle

•

C = C = 2d r

2A = r

Solid Figures and VolumesSolid Figures• cubes, rectangular solids, prisms,

cylinders, cones, spheres, and pyramids• volume of a rectangular solid (V= ) • volume of a right circular cylinder (V= )

• Recognize these solids

2r h

l w h

Surface Area• sum of areas of all the sides of the

solid• can use net to see sides of solid

Geometric PerceptionGeometric Perception Questions • require you to visualize a plane figure or a solid from different views or orientationsExample:

The wire frame above is made of three wires permanently joined together: a red wire, a blue wire, and a green wire. Three beads, labeled A, B, and C, are attached to the frame so that each of them can move all around the frame. However, none of the beads can be taken off the frame, nor can they be moved past one another. Which of the following configurations cannot be reached by sliding the beads around the frame or changing the position of the frame?

original figure

• The configuration in (A) can be reached by sliding each bead clockwise to the next wire piece.

• The configuration in (C) can be reached by sliding each bead counterclockwise to the next wire piece and then flipping the frame over.

• The configuration in (D) is reached simply by sliding bead A clockwise to the green wire.

• The configuration in (E) comes from turning the wire frame a third of a revolution clockwise.

• The configuration in (B) cannot be reached no matter how you slide the beads or rotate and flip the frame.

• The correct answer is (B).

Answer:

Coordinate Geometry

Parallel Lines• equal slopesPerpendicular Lines• product of slopes is -1 Positive Slope• Rises up left to right Negative Slope• Falls from left to right

2 3: 1

3 2ex

Midpoint• average of the coordinates•

Distance

•

1 2 1 2,2 2

x x y y

2 21 2 1 2( ) ( )d x x y y

TransformationsTranslation• moves a shape without any rotation or

reflection (up, down, left, right)Rotation• turning an object around a point, called

the center of rotation Reflection• mirror image with respect to a line,

which is called the line of reflection