Download - 1st Geometry Journal
1st Geometry Journal
By Daniel Escobar
What are points, lines, and segments?
• Point: A dot in space that indicates something or a location. Pic: .
• Line: A straight conection of dots that go for ever. Pic:
• Segment: A piece of line that has a begining and an end.
What are rays and Planes
• Plane: a flat surface Pic:• Rays: A conection of dots that have one
begining and go on for ever• How are a line, a segment, and a ray related to
each other?• 1. All of them make shapes.
What is the difference between a collinear and a coplanar point
• Collinear point: Points that lie in the same line.Collinear = line
• Coplanar Point: Points that lie in the same plane.
Coplanar = Plane
Noncollinear vs Noncoplanar
• Noncollinear: points not on the same line
• Noncoplanar
What is an intersection?
• An intersection is the set of all points that two or more figures have in common. My def: when two lines cross each other.
• Pic:
What is the difference between a postulate and an axiom, and theorem?
• Postulate/axiom is a statement that is accepted as true without proof.
• A Theorem is a statement that you can prove.• If you have proven a theorem, you can use it
as a reason in later proofs.
What is a Ruler Postulate
• Ex.1
• Ex.2
• Ex.3
1
A ruler postulate tells us that the points on a line can be paired on a one-to-one with a real number
7
13
2
12
11 14
6 9
What is a Segment Addition Postulate?
• The Segment Addition Postulate states that if B is between A and C then
AB + BC =AC
Ex.1 DF + FG = DG
Ex.2 GT + TE = GE
a b c
D FG
G T E
How to find the distance between two points on a coordinate plane?
• A coordinate Plane is a plane that is divided into four regions by a horizontal line (x-axis) and a verticle line (y-axis). The Distance between two points (x1, y1) and (x2, y2) is found with the following formula, the distance formula:
• d=√¯¯¯¯¯¯¯¯¯Ex.1 d=√¯¯¯¯¯¯¯¯ = √¯¯¯29Ex.2 d=√¯¯¯¯¯¯¯¯¯ =√¯¯¯5 .4Ex.3 d=√¯¯¯¯¯¯¯¯¯ =√¯¯¯15
(x2-x1)^2 + (y2-y1)^2
(1- -4)^2 + (-2-0)^2
(-4-1)^2 + (-4- -2)^2
(-2- -2)^2 + (-8-7)^2
Congruence vs. Equality
• When something is congruent, it is exactly the same/ same measure. Ex: AB≅CD
• When something is equal, it means it has the same value. Ex: 2=2
• They both compare two numbers/ solutions/products.
• Both relate to having two same products.
What Is the Pythagorean Theorem?
• The Pythagorean Thereom states that in a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. a2 + b2 = c2
6
4
62 +42 = 5252 = c2
√52= c7.2 = c
5
5
52 + 52 = 5050 = c2
√50= c7.07 = c
2
92 + 22 = 8585 = c2
√85= c9.2 = c
9
Angles
• An angle are two rays that share a common endpoint. An angle is measured by 3 points. Ex: ABC. The letter in the middle is the ∠vertex.
• There are 4 types of angles • Right angle: Measures 90°• Acute angle: Less than 90°• Obtuse angle: more than 90°• Straight Angle: Measures 180°
What is an Angle addition postulate?
• The Angle addition postulate states that if S is I the interior of PQS + m SQR = PQR.∠ ∠ ∠
Ex.1 m= JKM if m JKL= 42° and ∠ ∠m LKM=28°= 14°∠
Ex.2 m DEG=37° and m DEF=84° find m GEF ∠ ∠ ∠(84-37)= 47°= GEF∠
Ex.3 m LKM if m JKL =56.4° and m JKM ∠ ∠ ∠=82.5° (82.5- 56.4) = 26.1= m LKM∠
Midpoint
• A midpoint is a point that bisects, or divides a segment into 2 congruent segments. It can be found by dividing the measurements of the segments by two. Ex: AB=8 then AM=4, BM=4. AM=BM.
• CD= 5 CM=2.5, DM=2.5 • AB=6 AM=3, BM=3• AC, AB =2y and BC 8y-3• 2y= 8y-3 -8y -8y
-6y=-3 y=2
AB=4, BC= 13, AC = 17
Angle Bisector
• An Angle bisector is a ray that divides an angle into 2 congruent angles.
• To construct one you will need a compass. And follow the instructions on the picture below.
What are adjacent, vertical and linear pairs of angles??
• Adjacent angles: Two Angles that have the same vertex and share side.
• Linear Pairs: are two angles that create a straight line
• Vertical Angle: are two nonadjacent angles formed by two intersecting lines.
Complementary vs. Supplementary
• Complementary angles: are two angles that add up to 90°
• Supplementary angles: are any two angles that add up to 180°
52° 38°
55° 125°
How to find perimeter and area of te following shapes.
• Rectangle: P=2L +2w, A =Lw Ex: L=2cm w= 5cm (P=2 +25) P=27cm, (A=2x5) A=10 cm2
• Triangle: P= a+b+c, A= 1/2bh Ex: a=8, b=(x+1), c= 4x, and h=6 . P=5x +9, A= 3x+3
• Square: P= 4s, A=s2 Ex: 10 cm(P= 10x4) P=40 cm, (A=102) A=100 cm
How to find the area and circumference of a circle
• Area of a circle: A=(pie)2 Ex: 4cm4(pie)2 =16x(pie) ≈50.24 cm2
8cm ≈ 67.14 cm2
Circumference of a Circle: C =2(pie)r (R=radius:
a segment of a circle one of its endpoints are the center of the circle and another point on the circle.)
Ex: 4 cm C=8(pie) ≈25.12 C= 16(pie) ≈50.24
5 step process
• 1st Read and Analyze the Question• 2nd Find important info. And rewrite it• 3rd Visualize the information you just wrote• 4th Solve the equation• 5th Write the answer
Transformations
• Transformation: Change of position of an object.• Image A transformA A prime∆ABC ∆A’B’C’There are 3 types of transformations:TranslationReflection Rotation
Translation
• • • A
• B C• • A’
B’C’
Translation slides an object in any direction.
Reflection
• Reflection reflects/mirrors a figure across the line
If across the x axis (x,y) -> (x,-y). If across y axis (x,y) -> (-x,y)
prime
Rotation
• Rotation is when you rotate a figure around a point.
Prime
Thank you for your patience…