Transcript Geometry 2010 SOL

Modified and Animated By Chris Headlee
May 2010
SSM: Super Second-grader Methods
Lines and Angles
SSM:
• Use scratch paper to measure
AB
• See which line segment cuts AB
in half
Draw the connecting line segments with your ruler
See which one appears to cut AB in half
Lines and Angles
Alternate Interior
Alternate Interior angles are equal!
7x – 115 = 3x – 35
4x – 115 = -35
4x = 80
x = 20
SSM:
• both angle acute  EQUAL
• plug in answers to see which
make them equal
Lines and Angles
SSM:
• x is acute
ELIMINATE A and B
• Magic Number is 180
• x + what = 180
Angles must be formed by one transversal (line c)
x and 120 go together for a and b being parallel
x + 120 = 180
x = 60
Lines and Angles
SSM:
• Use straight-edge tool to
continue lines as far as possible
• Eliminate W and Z
• Compare distance between PY
and YR
• Compare distance between PX
and RX
Angle bisector:
1) draw equal distance points (arc tool) from Q
2) draw equal distance arcs from those two points
3) connect Q and two arc intersection point
4) Point Y is on the line (angle bisector)
Lines and Angles
SSM:
• Eliminate D (D is acute)
• D > B, eliminate A
angle A and D are alternate interior angles and must be the same
26 + 85 + mA = 180
mA = 180 – (26 + 85) = 180 – 111 = 69
Lines and Angles
SSM:
• Use scratch paper to measure BC
• Measure out from P
Use arc tool to set BC distance
Move tool to P and see that point R is answer
Lines and Angles
SSM:
• x is acute; eliminate C and D
• y is obtuse; eliminate A and C
• B is only answer left!
x and 30 have to be equal (alternate interior angles between l and n)
y and the vertical angle to 30 are consecutive interior angles; so 30 + y = 180
and y must be 150
Lines and Angles
SSM:
• 1 is acute; eliminate H and J
• 1 looks same as 32
32 and 1 are alternate interior angles
and so are equal
Lines and Angles
SSM:
• x is acute; eliminate C and D
• x is large acute; eliminate A
Parallel lines:
Alternate exterior angles are equal
Lines and Angles
SSM:
• 1 is obtuse; only answer J works
Vertical angles are equal
15x – 5 = 10x + 35
5x – 5 = 35
5x = 40
x=8
Substitute: m1 = 15(8) – 5 = 115
Lines and Angles
SSM:
• Our eyes tell us that x is obtuse
and w is acute
• acute + obtuse = 180 in parallel
lines
consecutive interior angles are supplementary;
w + 118 = 180
Triangles and Logic
SSM:
• 22 and 24 are “too big’ for the
picture of the ABC triangle
similar triangles: (order rules AC to LN)
10
(3x + 3)
---- = ----------5
(x + 5)
10(x + 5) = 5(3x + 3)
10x + 50 = 15x + 15
50 = 5x
35 = x
7=x
AC = 7 + 5 = 12
Triangles and Logic
SSM:
• no common p-triples (no help)
Pythagorean Thrm:
9² + 40² = 41²
81 + 1600 = 1681
1681 = 1681
others do not satisfy Pythagorean Theorem
Triangles and Logic
SSM:
• intersection  both true
Everything in rectangle are quadrilaterals  answer H is wrong
Trapezoids are a quadrilaterals  answer J is wrong
Since the two circles do not completely overlap  answer F is wrong
Triangles and Logic
SSM:
• bisect cuts sides in half  two S’s;
eliminates A and B
• Need another side or an angle
Bisects means AE and EC are congruent
and BE and ED are congruent
vertical angles; AED  CEB
this yields SAS (since angle is included)
Triangles and Logic
SSM:
• no help
Contrapositive is a flip and a negate
flip changes order of statements and negate adds a not to both
(remember a double negative is a positive!)
Triangles and Logic
SSM:
• plot all answers
• see which one makes sense
(triangles look the same)
AB and CD must match up (order rules!)
A to B is down 1 and over 4
From C to D must be over 4 and up 1
Triangles and Logic
SSM:
• find the nots (~) and see which
answer fits
• G and J have no nots; eliminate
If an equation is of the form y = mx + b, then its graph is a line.
pq
The graph is not a line
~q
Therefore, the equation is not of the form y = mx + b
 ~p
Triangles and Logic
SSM:
• x < 22 ; eliminates D
• answer A doesn’t form a triangle
Pythagorean Theorem:
x² + 17² = 22²
x² + 289 = 484
x² = 195
x = 13.96
Triangles and Logic
SSM:
• measure sides with
scrap paper
• AC is shortest
• BC is longest
Order the measures of angles from largest to smallest:
Replace with the letter of the angles
Put in the missing letters of the triangle
Now the sides are ordered from longest to shortest:
75 >
A >
BC >
BC >
55
C
AB
AB
> 50
> B
> AC
> BD
Triangles and Logic
SSM:
• largest angle opposite largest side
Order the measures of sides from smallest to largest:
Replace with the letters of the sides
Put in the missing letters of the triangle
Now the angles are ordered from smallest to largest:
150
HI
J
J
<
<
<
<
245
HJ
I
I
< 365
< IJ
< H
< H
Triangles and Logic
SSM:
• answer A is wrong; x
must be bigger than 6
Pythagorean Thrm;
45-45-90 triangle (isosceles)
6² + 6² = x²
36 + 36 = x²
72 = x²
8.49 = x (62)
Special Case right triangles:
side opposite 45 angle is ½ hyp 2
so 6 = ½ x 2
12 = x 2
62 = x
Trig:
6 is O; x is H; use sin
sin 45 = 6 / x
x = 6 / (sin 45) = 8.49
Triangles and Logic
SSM:
• any two sides bigger than 3rd
third side must:
14 – 8 < 3rd side < 14 + 8
6 < 3rd side < 22
all answers except A fit inequality above
Polygons and Circles
SSM:
• H and J are obtuse and ABC
is acute
• ABC is small acute so F is
better choice
Included angle = ½ m arc
= ½ (60) = 30
Polygons and Circles
SSM:
• x is acute so C and D are wrong.
• x is smaller acute (compared to other
acute angle in )
Regular hexagon: each angle is [ (n – 2)180 = 720 ] 6 = 120
120 is an exterior angle to triangle  so 90 + x = 120
x = 30
Polygons and Circles
SSM:
• Look to find examples of each figure
• Rectangles and squares have 4 – 90
angles
• One side of a trapezoid is shorter or
longer than the others
Squares and Rhombus are only quadrilaterals with four equal sides.
Squares have all 90 angles.
Polygons and Circles
SSM:
• DAR is bigger than RBD
• Eliminates A and B
RBD is included and = ½ m arc RD
RAD is central and = m arc RD
so 2(34) = 68 = RAD
Polygons and Circles
SSM:
• by measuring x is very close to 7.5
512 = (7.5) x
60 = 7.5x
8=x
Polygons and Circles
SSM:
• look at the opposite sides of a rectangle
• They are parallel!
Rectangles are parallelograms
Parallelograms opposite sides are parallel
parallel lines have the same slope
Polygons and Circles
SSM:
• WXY is obtuse; eliminates F
• 180 is magic number
WXY is an exterior angle and equal to sum of remote interior
WXY = 62 + 73 = 135
Polygons and Circles
SSM:
• compare EFG to GDE
• they are the same!
rhombus is a parallelogram
opposite angles in parallelogram are equal
Polygons and Circles
SSM:
• x is an acute angle; eliminate
H and J
• fold corner of paper in half to
compare to 45  equal!
x is an exterior angle of an octagon (8 sides)
8x = 360 (sum of exterior angles = 360)
x = 45
Polygons and Circles
SSM:
• Numbers in C and D don’t fit
inside the sides of the rectangle
Drawing a picture gives us a right-triangle in a corner 
so Pythagorean Thrm applies
midpoints divide rectangle sides in half!
5² + 12² = x²
25 + 144 = x²
169 = x²
13 = x
Three-Dimensional Figures
SSM:
• man’s shadow is smaller
than the man
• flagpole’s shadow must be
smaller than flagpole
• Eliminates F and G
Similar Triangle problem:
6
-4
h
= ---18

4h = 108
h = 27
Three-Dimensional Figures
SSM:
• Find formula (rectangular prism)
• find variables
• plug in and solve
V = lwh
= 201012 = 2400 cu in (when full)
V = (4/5)(2400) = 1,920 cu in
Three-Dimensional Figures
fold them up in your mind
no gaps or overlaps
SSM:
• Label each part either
F(front)
Bk (back)
S (side)
T (top)
B (bottom)
• all but one that has a missing ltr
Three-Dimensional Figures
SSM:
• Find formula (V = s³)
• find variables
• plug in Volume and solve for sides
• compare answers
V=lwh
= 3 3  3
= 27 cu in
(3 = ½ 6)
1/8 times original volume
Three-Dimensional Figures
SSM:
• see which one looks false
bottom and top views are circles
front (and side) view is a triangle
leaves J as incorrect
Three-Dimensional Figures
SSM:
• use formula sheet
• solve backward for r and
see which makes sense
LA = rl
rl = 60
10r = 60
r=6
V = (4/3)r³ = (4/3)(6)³ = 288
Coordinate Relations and Transformations
SSM:
• plot the points and the lines
of reflection
• see which is equal distant
plot points and then the lines of reflection
y
x
Coordinate Relations and Transformations
SSM:
• plot points and see
which make sense
Midpoint formula:
[(x1 + x2)/2 , (y1 + y2)/2] = [(1 + x)/2 , (1 + y)/2] = [-2, 0]
(1 + x)/2 = -2  1 + x = -4
x = -5
(1 + y)/2 = 0  1 + y = 0
y = -1
Coordinate Relations and Transformations
SSM:
• slide
Triangle slid over  Translation
Coordinate Relations and Transformations
SSM:
• fold over x-axis
• y value switches sign
symmetric to x-axis is (-1)  y-value
Coordinate Relations and Transformations
Midpoint formula:
Use to find each mid-point
SSM:
• plot the answers (points)
• see which point corresponds to intersection
Coordinate Relations and Transformations
SSM:
• draw figure
• draw lines of symmetry
Regular quadrilateral is a square and has four lines of symmetry
Even numbered regular polygons have a point of symmetry