Download ppt-solutions-to-2016-2017-geometry-midterm-review-1

Rectangle ABCD is shown below. Find the midpoint of
diagonal 𝐴𝐶.
(𝟒, 𝟒)
(−𝟑, 𝟏)
(𝟎. 𝟓, 𝟐. 𝟓)
𝑥1 + 𝑥2 𝑦2 + 𝑦2
𝑚=
,
2
2
−3 + 4
,
2
1
= 0.5,
2
1+4
2
5
= 2.5
2
Use the distance formula…
𝑑=
(𝑥1 − 𝑥2 )2 +(𝑦1 − 𝑦2 )2
𝑑=
(7 − 1)2 +(4 − 2)2
𝑑=
(6)2 +(2)2
𝑑 = 36 + 4
𝑑 = 40
Use the midpoint formula…
𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 =
𝑥1 +𝑥2 𝑦1 +𝑦2
, 2
2
𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 =
8 6
, = (4, 3)
2 2
7+1 4+2
,
2
2
𝑑 = 2 10 ≈ 6.3
Use the midpoint formula…
𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 =
𝑥1 +𝑥2 𝑦1 +𝑦2
, 2
2
𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 =
Use the distance formula…
𝑑=
(𝑥1 − 𝑥2 )2 +(𝑦1 − 𝑦2 )2
𝑑=
(9 − 7)2 +(10 − 6)2
𝑑=
(2)2 +(4)2
𝑑 = 4 + 16
𝑑 = 20
𝑑 = 2 5 ≈ 4.47
9+7 10+6
,
2
2
16 16
= (8, 8)
,
2 2
Use the midpoint formula…
𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 =
Use the distance formula…
𝑑=
(𝑥1 − 𝑥2 )2 +(𝑦1 − 𝑦2 )2
7=
8+𝑥
,
2
𝑥1 +𝑥2 𝑦1 +𝑦2
, 2
2
3=
4+𝑦
2
14 = 8 + 𝑥; 6 = 4 + 𝑦
𝑑=
(8 − 6)2 +(4 − 2)2
6 = 𝑥; 2 = 𝑦
𝑑=
(2)2 +(2)2
(6, 2)
𝑑 = 4+4
𝑑= 8
𝑑 = 2 2 ≈ 2.828
Clear the denominator by
multiplying each by two…
6
∟
8 + 6 = 14 𝑏𝑙𝑜𝑐𝑘𝑠
8
Direct path
10 𝑏𝑙𝑜𝑐𝑘𝑠
Use the Pythagorean
Theorem
𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐
6 2 + 82 = 𝑐 2
36 + 64 = 𝑐 2
100 = 𝑐 2
10 = 𝑐
14 𝑏𝑙𝑜𝑐𝑘𝑠 − 10 𝑏𝑙𝑜𝑐𝑘𝑠 = 4 𝑏𝑙𝑜𝑐𝑘𝑠 𝑠ℎ𝑜𝑟𝑡𝑒𝑟
𝑑=
(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2
𝑑=
(12 − 4)2 + (6 − 0)2
𝑑=
(8)2 + (6)2
Use the distance
formula to find the
distance on a
coordinate plane.
𝑑 = 64 + 36
𝑑 = 100
𝑑 = 10
10 × 0.75 = 7.5 miles
Find the distance of
𝑨𝑩 for both 𝒙 and 𝒚
coordinates…
16 − 1 = 15 𝑎𝑛𝑑 8 − 3 = 5
𝟐
Find
𝟓
of each
coordinate distance…
2
2
× 15 = 6 𝑎𝑛𝑑 × 5 = 2
5
5
Since the point is partitioned
from point A, subtract from
point A’s coordinates…
16 − 6 = 10 𝑎𝑛𝑑 8 − 2 = 6
(𝟏𝟎, 𝟔)
2 parts
5 parts
(15.5, 0)
3
7.5
8 + 7.5 = 15.5
3 ÷ 2 𝑝𝑎𝑟𝑡𝑠 = 1.5
5 𝑝𝑎𝑟𝑡𝑠 × 1.5 = 7.5
Find the distance of 𝑨𝑩
for both 𝒙 and 𝒚
coordinates…
| − 9 − 12| = 21 𝑎𝑛𝑑 |2 − 8| = 6
𝟏
Find
𝟑
of each
coordinate distance…
1
1
× 21 = 7 𝑎𝑛𝑑 × 6 = 2
3
3
Since the point is partitioned
from point A, add to point
A’s coordinates…
−9 + 7 = −2 𝑎𝑛𝑑 2 + 2 = 4
(−𝟐, 𝟒)
Find the slope of the
line with points (4, 9)
and (-3, 6)…
𝑦2 − 𝑦1
𝑚=
𝑥2 − 𝑥1
Use the Point-Slope
Formula
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )
Simplify
6−9
−3
3
𝑚=
=
𝑜𝑟
−3 − 4 −7
7
Add 8 to each
side of the
equation…
3
𝑦 − 8 = (𝑥 − 14)
7
3
𝑦−8= 𝑥−6
+8 7 +8
3
𝑦 = 𝑥+2
7
Use the Point-Slope
Formula
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )
Find the slope of the
line with points (4, 9)
and (-3, 6)…
𝑦2 − 𝑦1
𝑚=
𝑥2 − 𝑥1
Simplify
6−9
7
−3
3
𝑚=
=
𝑜𝑟 = −
−3 − 4 −7
3
7
Perpendicular
slopes are
opposite sign
reciprocals.
7
𝑦 − 8 = − (𝑥 − 14)
3
7
98
𝑦−8=− 𝑥+
3
3
Add 8 to each
24
side of the
+8
+8
+
equation…
3
7
122
8 3 24 Simplify 𝑦 = − 𝑥 +
3
3
× =
1 3
3
7
2
𝑦 = − 𝑥 + 40
3
3
Use the Point-Slope
Formula
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )
𝑦 = 3𝑥 − 5
The slope of a
parallel line would
be +𝟑
Standard Form
𝐴𝑥 + 𝐵𝑦 = 𝐶
Simplify
Subtract 3x from
each side of the
equation…
Add 5 to each side
of the equation…
𝑦 − 5 = 3(𝑥 − 8)
𝑦 − 5 = 3𝑥 − 24
−3𝑥
−3𝑥
−3𝑥 + 𝑦 − 5 = −24
+5
+5
−3𝑥 + 𝑦 = −19
3𝑥 − 𝑦 = 19
OR
Use the Point-Slope Formula
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )
The slope of a
1
perpendicular line
𝑦 − 5 = − (𝑥 − 8)
3
𝟏
Simplify
would be − .
𝟑
1
8
𝑦−5=− 𝑥+
3
3
Add 5 to each side
Standard Form
15
of the equation…
+5
+5
𝐴𝑥 + 𝐵𝑦 = 𝐶
+
5 3 15
3
× =
1
23
1 3
3
𝑦=− 𝑥+
3
3
𝟏
Add 𝟑 𝒙 to each side
1
1
of the equation…
+ 𝑥
+ 𝑥
3
3
Multiply each term (3) 1
(3) 23 (3)
by 3…
𝑥+𝑦 =
3
3
𝑥 + 3𝑦 = 23
𝑦 = 3𝑥 − 5
𝑎.
𝑏.
𝒚 = −𝟓𝒙 − 𝟑
𝒚 = −𝟓𝒙 − 𝟑
The slope of a
parallel line would
be −𝟓.
y = −5𝑥 + 6
The slope of a
perpendicular line
𝟏
would be + .
𝟓
1
y= 𝑥+6
5
Translate 𝐴𝐵 using the translation rule 𝑥, 𝑦 → (𝑥 − 2, 𝑦 + 4) then
translate 𝐴′𝐵′ using the translation rule 𝑥, 𝑦 → 𝑥 − 2, 𝑦 − 6 ..
𝑨′
Translate 2 units left…
Translate 4 units up…
𝑨"
𝑩′
Translate 2 units left…
Translate 6 units down.
𝑩"
𝑩"
𝑨′
𝑨"
𝑩′
Rotate 𝐴𝐵 90° counterclockwise around the origin.
𝑅90° → (−𝑦, 𝑥)
𝑩′
𝑨′
(𝟑, 𝟏)
𝑨′ = (−𝟏, 𝟑)
𝑩′ = (𝟒, 𝟒)
(𝟒, −𝟒)
Rotate 𝐴𝐵 90° clockwise around the origin.
𝑅−90° → (𝑦, −𝑥)
(𝟑, 𝟏)
𝑨′ = (𝟏, −𝟑)
𝑩′ = (−𝟒, −𝟒)
𝑨′
𝑩′
(𝟒, −𝟒)
a. What is the scale factor of the dilation of line segment 𝐴𝐵 ?
The scale factor
would be 2.
6
3
𝑨𝑩 𝐢𝐬 𝐭𝐡𝐞 𝐩𝐫𝐞𝐢𝐦𝐚𝐠𝐞 𝐚𝐧𝐝
𝐡𝐚𝐬 𝐚 𝐥𝐞𝐧𝐠𝐭𝐡 𝐨𝐟 𝟑 𝐮𝐧𝐢𝐭𝐬.
b. Draw a segment demonstrating a dilation of 𝐴𝐵 by a
1
3
factor of .
𝑩′
𝑨′
Use a compass and straight edge to copy angle A.
(𝟕𝒙 − 𝟐𝟒)°
2
𝑚∠𝐿𝑀𝑁 = 176 °
3
(𝟐𝒙 + 𝟑𝟏)° (𝟐𝒙 + 𝟑𝟏)°
1
1
88 ° 88 °
3
3
Combine Like Terms
2
86
+ 31
3
Since 𝑴𝑶 is an angle
bisector, 𝒎∠𝑳𝑴𝑶 is
also 𝟐𝒙 + 𝟑𝟏.
Use the Angle Addition
Postulate
2𝑥 + 31 + 2𝑥 + 31 = 7𝑥 − 24
4𝑥 + 62 = 7𝑥 − 24
Subtract 4x from each side
−4𝑥
−4𝑥
of the equation…
62 = 3𝑥 − 24
Add 24 to each side of the
+24
+24
172
equation…
Divide each side of the
+ 31
86 = 3𝑥
3
equation by 3…
3
3
1
172 93 265
𝟖𝟔
Substitute for x to find the
= 88
+
=
𝟑
86
3
3
3
3
=𝑥
𝒎∠𝑳𝑴𝑶 𝐀𝐍𝐃 𝒎∠𝑵𝑴𝑶 …
3
Vertical Angle Theorem
8𝑥 − 9 = 7𝑥 + 7
−7𝑥
−7𝑥
𝑥−9=7
+9 +9
𝑥 = 16
7 16 + 7 = 119°
Substitute
Vertical Angle Theorem
6𝑥 − 9 = 2𝑥 + 7
−2𝑥
−2𝑥
4𝑥 − 9 = 7
+9 +9
4𝑥 = 16
4
4
𝑥=4
2 4 + 7 = 15°
Linear Pair
144°
144°
36°
(two angles that share
a common ray that
form a straight line)
(8𝑥 − 64)° + 36° = 180°
8 26 − 64 = 144°
Add 28 to
each side of
the equation…
8𝑥 − 28 = 180
+28 +28
Divide each side
of the equation
by 8…
8𝑥 = 208
8
8
𝑥 = 26
Combine Like
Terms…
Linear Pair
135°
135°
45°
(two angles that share
a common ray that
form a straight line)
(5𝑥 − 20)° + 45° = 180°
5 31 − 20 = 135°
Combine Like
Terms…
Subtract 25 from
each side of the
equation…
Divide each side
of the equation
by 5…
5𝑥 + 25 = 180
−25 −25
5𝑥 = 155
5
5
𝑥 = 31
𝑎 ∥𝑏
𝑨𝒍𝒕. 𝑬𝒙𝒕. ∠𝟏 & ∠𝟓, ∠𝟐 & ∠𝟔
𝑨𝒍𝒕. 𝑰𝒏𝒕. ∠𝟒 & ∠𝟖, ∠𝟑 & ∠𝟕
𝑺𝒂𝒎𝒆 − 𝑺𝒊𝒅𝒆 𝑰𝒏𝒕. ∠𝟖 & ∠𝟕, ∠𝟑 & ∠𝟒
𝑪𝒐𝒓𝒓𝒆𝒔𝒑𝒐𝒏𝒅𝒊𝒏𝒈 ∠𝟏 & ∠𝟕, ∠𝟐 &∠𝟒, ∠𝟑 & ∠𝟓, ∠𝟖 & ∠𝟔
𝑎 ∥𝑏
𝑪𝒐𝒓𝒓𝒆𝒔𝒑𝒐𝒏𝒅𝒊𝒏𝒈 𝑨𝒏𝒈𝒍𝒆𝒔: ∠𝟏 ≅ ∠𝟕; ∠𝟐 ≅ ∠𝟒; ∠𝟑 ≅ ∠𝟓; ∠𝟔 ≅ ∠𝟖
𝑨𝒍𝒕. 𝑰𝒏𝒕𝒆𝒓𝒊𝒐𝒓 𝑨𝒏𝒈𝒍𝒆𝒔: ∠𝟑 ≅ ∠𝟕; ∠𝟒 ≅ ∠𝟖
𝑨𝒍𝒕. 𝑬𝒙𝒕𝒆𝒓𝒊𝒐𝒓 𝑨𝒏𝒈𝒍𝒆𝒔: ∠𝟏 ≅ ∠𝟓; ∠𝟐 ≅ ∠𝟔
𝑺𝒂𝒎𝒆 − 𝑺𝒊𝒅𝒆 𝑰𝒏𝒕𝒆𝒓𝒊𝒐𝒓 𝑨𝒏𝒈𝒍𝒆𝒔 𝒂𝒓𝒆 𝑺𝒖𝒑𝒑𝒍𝒆𝒎𝒆𝒏𝒂𝒓𝒕𝒚: ∠𝟑 + ∠𝟒; ∠𝟕 + ∠𝟖
∠𝟔 ≅ ∠𝟐 as they are
corresponding angles.
∠𝟏 = 𝟏𝟏𝟐° by
alternate
exterior angles
𝟏𝟏𝟐°
𝟏𝟏𝟐°
𝟏𝟏𝟐°
∠𝟒 = 𝟏𝟏𝟐° by
vertical angles
112° + 112° = 𝟐𝟐𝟒°
Vertical
Angles
𝟕𝟖°
Alt. Interior
Angles
Vertical
Angles
180° − 40° − 62° = 78°
𝟒𝟎° 𝟕𝟖°
𝟒𝟎°
3 angles that form
a straight line
𝟏𝟏𝟖°
Remote Exterior
Angle
78° + 40° = 118°
𝟕𝟖°
𝟒𝟎°
𝟒𝟎°
𝟕𝟖°
𝟏𝟏𝟖°
𝟕𝟖°
Alt. Interior
Angles
Vertical
Angles
𝟒𝟎°
𝟕𝟖°
3 angles that form
𝟒𝟎° a straight line
180° − 40° − 62° = 78°
Complete the table below.
𝟏𝟖𝟎 𝒏 − 𝟐 𝐰𝐡𝐞𝐫𝐞 𝒏 𝐢𝐬
𝐭𝐡𝐞 𝐧𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐬𝐢𝐝𝐞𝐬
𝐨𝐟 𝐭𝐡𝐞 𝐩𝐨𝐥𝐲𝐠𝐨𝐧.
𝟏𝟖𝟎 𝒏 − 𝟐
𝐰𝐡𝐞𝐫𝐞 𝒏 𝐢𝐬
𝒏
𝐭𝐡𝐞 𝐧𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐬𝐢𝐝𝐞𝐬
𝐨𝐟 𝐭𝐡𝐞 𝐩𝐨𝐥𝐲𝐠𝐨𝐧.
𝟑𝟔𝟎
𝐰𝐡𝐞𝐫𝐞 𝒏 𝐢𝐬
𝒏
𝐭𝐡𝐞 𝐧𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐬𝐢𝐝𝐞𝐬
𝐨𝐟 𝐭𝐡𝐞 𝐩𝐨𝐥𝐲𝐠𝐨𝐧.
− 𝟐)
𝟏𝟖𝟎(𝟑)
𝟏𝟎𝟖°
PENTAGON 𝟏𝟖𝟎(𝟓
𝟏𝟖𝟎(𝟑)
− 𝟐) 𝟏𝟖𝟎(𝟓
𝟓𝟒𝟎°
𝟓
𝟑𝟔𝟎
𝟕𝟐°
𝟓
− 𝟐)
𝟏𝟖𝟎(𝟔)
− 𝟐) 𝟏𝟖𝟎(𝟖
OCTAGON 𝟏𝟖𝟎(𝟖
𝟏𝟑𝟓°
𝟏𝟖𝟎(𝟔)
𝟏𝟎𝟖𝟎°
𝟖
𝟑𝟔𝟎
𝟒𝟓°
𝟖
𝟏𝟖𝟎(𝟓)
𝟏𝟐𝟖.−𝟓°𝟐)
𝟏𝟖𝟎(𝟓)
− 𝟐) 𝟏𝟖𝟎(𝟕
HEPTAGON 𝟏𝟖𝟎(𝟕
𝟗𝟎𝟎°
𝟕
𝟑𝟔𝟎𝟓°
𝟓𝟏.
𝟕
𝟏𝟖𝟎(𝟒)
− 𝟐)
𝟏𝟐𝟎°
𝟏𝟖𝟎(𝟒)
𝟕𝟐𝟎°− 𝟐) 𝟏𝟖𝟎(𝟔
HEXAGON 𝟏𝟖𝟎(𝟔
𝟔𝟔
𝟑𝟔𝟎
𝟔𝟎°
𝟔
𝟏𝟖𝟎(𝟑
𝟏𝟖𝟎(𝟏)
𝟔𝟎°− 𝟐)
𝟑
𝟑𝟔𝟎
𝟏𝟐𝟎°
𝟑
TRIANGLE
𝟏𝟖𝟎(𝟑
𝟏𝟖𝟎(𝟏)
𝟏𝟖𝟎°− 𝟐)
𝟏𝟖𝟎(𝟖)
𝟏𝟒𝟒°− 𝟐)
𝟏𝟒𝟒𝟎°− 𝟐)𝟏𝟖𝟎(𝟏𝟎
𝟏𝟖𝟎(𝟖)
DECAGON 𝟏𝟖𝟎(𝟏𝟎
𝟏𝟎
𝟑𝟔𝟎
𝟑𝟔°
𝟏𝟎
What is the measure of the missing angle in the polygons below?
𝐒𝐮𝐦 𝐨𝐟 𝐭𝐡𝐞 𝐈𝐧𝐭𝐞𝐫𝐢𝐨𝐫 𝐀𝐧𝐠𝐥𝐞𝐬
𝟏𝟖𝟎(𝒏 − 𝟐)
180(5 − 2)
180(8 − 2)
180(3) = 540°
180(6) = 1080°
540° − 90 − 145 − 95 − 165 = 45°
1080° − 150 − 120 − 112 − 108 − 170 − 165 − 150 = 105°
B’
A’
2
3
C’
Translate 5 units left…
B’
A’
C’
Translate 2 units down.
A’
C’
4
4
1 B’
1
4
4
B’
C’
A’
𝑪′
𝑩′
𝑨′
Rotations without
specification are
always
counterclockwise!
The distance from
the pre-image point
to the origin is the
same distance as the
image point to the
origin. Note the 90
degree angle formed.
The distance from
the pre-image point
to the origin is the
same distance as the
image point to the
origin. Note the 180
degree angle formed.
Reflect across the
y-axis.
𝐶′
𝐶′′
𝐴′
𝐵′
𝐴′′
𝟏𝟖𝟎°
𝟏𝟖𝟎°
𝟏𝟖𝟎°
𝐵′′
𝑪"
𝑪′
𝑨′
𝑩′
𝑨"
𝑩"
𝑅180° → (−𝑥, −𝑦)
What is the scale factor in the dilation shown below?
Pre Image
Since the image
is getting smaller,
the scale factor
will be fraction…
4 units
2 units
The scale factor will be ½.
What is the scale factor in the dilation shown below?
Pre Image
Since the image is
getting larger, the
scale factor will be
greater than 1…
3 units
6 units
The image is twice as large as the pre
image, so the scale factor would be 2.
Draw the translated image (𝑥, 𝑦) → (𝑥 + 6, 𝑦 − 1) and then
reflect the image over the x-axis.
Translate triangle
ABC 6 units right
and 1 unit down.
𝐵′
𝐴′
𝐴"
𝐵"
𝐶′
𝐶"
Reflect triangle
A’B’C’ over the xaxis.
By the Linear Pair
Postulate…
𝟏𝟖𝟎° − 𝟏𝟏𝟔° = 𝟔𝟒°
𝟔𝟒°
By the Triangle Angle
Sum Theorem…
𝟏𝟖𝟎° − 𝟕𝟓° − 𝟔𝟒° = 𝟒𝟏°
By the Linear Pair
Postulate…
𝟏𝟖𝟎° − 𝟏𝟑𝟏° = 𝟒𝟗°
𝟒𝟗°
By the Triangle Angle
Sum Theorem
(the sum of the interior
angles of a triangle is
equal to 𝟏𝟖𝟎°)…
𝟏𝟖𝟎° − 𝟔𝟏° − 𝟒𝟗° = 𝟕𝟎°
A softball home plate has been designed on the coordinate
plane shown below. If each unit is 2 inches, what is the area of
home plate?
1
𝐴 = (𝑙 × 𝑤)
Divide the shape
2
into a rectangle
1
and a triangle…
𝐴 = 6×3 =9
2
𝟗 𝒔𝒒. 𝒖𝒏𝒊𝒕𝒔
𝐴 =𝑙×𝑤
𝐴 = 6 × 4 = 24
𝟐𝟒 𝒔𝒒. 𝒖𝒏𝒊𝒕𝒔
𝟐𝟒 + 𝟗 = 𝟑𝟑 𝒔𝒒. 𝒖𝒏𝒊𝒕𝒔
𝟑𝟑 𝒔𝒒. 𝒖𝒏𝒊𝒕𝒔 × 𝟐𝟐 = 𝟏𝟑𝟐 𝒔𝒒. 𝒊𝒏𝒄𝒉𝒆𝒔
If each unit in the coordinate plane below is 3 cm, what is the
area of the polygon ABCDEF?
Divide the shape
into a rectangle
and 2 triangles…
𝟏𝟔 𝒔𝒒. 𝒖𝒏𝒊𝒕𝒔
1
𝐴 = (𝑙 × 𝑤)
2
1
𝐴 = 8 × 4 = 16
2
𝐴 =𝑙×𝑤
𝐴 = 8 × 3 = 24
𝟐𝟒 𝒔𝒒. 𝒖𝒏𝒊𝒕𝒔
𝟏𝟐 𝒔𝒒. 𝒖𝒏𝒊𝒕𝒔
1
𝐴 = 8 × 3 = 12
2
𝟐𝟒 + 𝟏𝟔 + 𝟏𝟐 = 𝟓𝟐 𝒔𝒒. 𝒖𝒏𝒊𝒕𝒔
𝟓𝟐 𝒔𝒒. 𝒖𝒏𝒊𝒕𝒔 × 𝟑𝟐 = 𝟒𝟔𝟖 𝒔𝒒 𝒄𝒎
Reflexive Property
Symmetric Property
Transitive Property
27°
𝟏𝟑
𝟏𝟐
𝟔𝟕° 𝟗𝟎°
𝟓
If the triangles are congruent, then
their corresponding parts are
congruent. (CPCTC)
What other information do you need in order to prove the triangles congruent
using the SAS Congruence Postulate? By AAS Congruence Theorem?
Hidden is the fact that
𝑨𝑪 ≅ 𝑨𝑪 by the
reflexive property.
𝑨
𝑺
𝑺
The missing
information is
𝑨𝑩 ≅ 𝑨𝑫 so that the
triangle are congruent
by SAS.
The missing
information is
∠𝑩 ≅ ∠𝑫 so that the
triangle are congruent
by AAS.
𝑨
𝑨
𝑺
𝑨
𝑺
𝑺
Hidden is the fact that
𝑨𝑪 ≅ 𝑨𝑪 by the
reflexive property.
𝑨
∠𝐵𝐴𝐶 ≅ ∠𝐷𝐶𝐴 𝑓𝑜𝑟 𝐴𝑆𝐴
𝐴𝐷 ≅ 𝐵𝐶 𝑓𝑜𝑟 𝑆𝐴𝑆
AB is congruent to DC and AC is congruent to DB
by given information. By the reflexive property,
BC is congruent to BC. Triangle ABC is congruent
to triangle DCB by the side, side, side postulate.
𝐴𝐶 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠𝐷𝐴𝐵, 𝐶𝐴 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠𝐷𝐶𝐵
𝐺𝑖𝑣𝑒𝑛
∠𝐷𝐴𝐶 ≅ ∠𝐵𝐴𝐶, ∠𝐷𝐶𝐴 ≅ ∠𝐵𝐶𝐴
𝐷𝑒𝑓 𝑜𝑓 𝑎𝑛𝑔𝑙𝑒 𝑏𝑖𝑠𝑒𝑐𝑡𝑜𝑟
𝐴𝐶 ≅ 𝐴𝐶
𝑅𝑒𝑓𝑙𝑒𝑥𝑖𝑣𝑒 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦
∆𝐷𝐴𝐶 ≅ ∆𝐵𝐴𝐶
𝐴𝑆𝐴
𝑺
𝑺
𝑨
𝐴𝐵 ≅ 𝐴𝐶 𝑎𝑛𝑑 ∠𝐵𝐴𝐷 ≅ ∠𝐶𝐴𝐷
𝐺𝑖𝑣𝑒𝑛
𝐴𝐷 ≅ 𝐴𝐷
𝑅𝑒𝑓𝑙𝑒𝑥𝑖𝑣𝑒 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦
∆𝐴𝐶𝐷 ≅ ∆𝐴𝐵𝐷
𝑆𝐴𝑆
𝐶𝐷 ≅ 𝐵𝐷
𝐶𝑃𝐶𝑇𝐶
𝐴𝐷 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 𝐵𝐶
𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑠𝑒𝑔𝑚𝑒𝑛𝑡 𝑏𝑖𝑠𝑒𝑐𝑡𝑜𝑟
𝑨𝑨~
∠𝑨𝑫𝑬 ≅ ∠𝑨𝑩𝑪 𝒃𝒚 𝑪𝒐𝒓𝒓𝒆𝒔𝒑𝒐𝒏𝒅𝒊𝒏𝒈 𝑨𝒏𝒈𝒍𝒆 𝑷𝒐𝒔𝒕𝒖𝒍𝒂𝒕𝒆
∠𝑫𝑨𝑬 ≅ ∠𝑩𝑨𝑪 𝒃𝒚 𝒕𝒉𝒆 𝑹𝒆𝒇𝒍𝒆𝒙𝒊𝒗𝒆 𝑷𝒓𝒐𝒑𝒆𝒓𝒕𝒚
In the diagram below, what is the length of 𝑆𝑇?
By the Triangle
Proportionality
Theorem…
𝟓 𝟏𝟎
=
𝟒
𝒙
Divide each side of
the equation by 5…
5𝑥 = 40
5
5
𝑥=8
Cross Multiply
The question is
asking for ST not
the value of x!!!
𝑆𝑇 = 10 + 8 = 18
By the Triangle
Proportionality
Theorem…
𝒙
𝒙−𝟐
=
𝒙+𝟓 𝒙+𝟏
Divide each side
of the equation
by -2…
Cross Multiply
𝑥 2 + 𝑥 = 𝑥 2 + 3𝑥 − 10
𝑥 = +3𝑥 − 10
−3𝑥 −3𝑥
−2𝑥 = −10
−2
−2
𝑥=5
Subtract 3x
from each side
of the
equation…
∆𝐵𝐷𝐹 = 25 + 35 + 21 = 81
𝟒𝟐
𝟐𝟓
𝟕𝟎
𝟐𝟏
𝟑𝟓
𝟓𝟎
∆𝐴𝐶𝐸 = 42 + 50 + 70 = 162
Triangle Midsegment
(𝟏 𝟐 the length of the parallel side)
By the triangle Midsegment
Theorem…
Simplify…
Add 8 to each
side of the
equation…
Triangle Midsegment
(𝟏
𝟐
the length of the parallel side)
2 2𝑥 − 4 = 20
4𝑥 − 8 = 20
+8 +8
4𝑥 = 28
Divide each
4
4
side of the
equation by 4…
𝑥=7
136 𝑓𝑡.
The tic marks
indicate that this is
the midsegment of
the triangle.
By the Triangle Midsegment
Theorem, the midsegment is
½ the length of its parallel side
(or the parallel side is double the
length of the midsegment)…
68 × 2 = 136 𝑓𝑡.
The smallest side must be
greater than the difference of
the two other sides…
The largest side must be less
than the sum of the two
other sides…
14 − 7 = 7
14 + 7 = 21
𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟: 𝐺𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 7 𝑎𝑛𝑑 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 21
Two sides of a triangle have lengths 3 and 9. What must be true about the
length of the third side?
The smallest side must be
greater than the difference of
the two other sides…
9−3=3
The largest side must be less
than the sum of the two
other sides…
9 + 3 = 12
𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟: 𝐺𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 3 𝑎𝑛𝑑 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 12
<
98°
180° − 82° = 98°
By the Hinge Theorem, Since
98° > 82°, 𝐾𝐿 > 𝐿𝑀.
A pair of scissors is opened to two positions, A and B.
<
By the Hinge Theorem, Since
23° > 29°, 𝐶𝐷 > 𝐴𝐵.
The Centroid patricians
the medians in a ratio of
2 to 1.
𝑨𝑬 = 𝟏𝟖 + 𝟗 𝒐𝒓 𝟐𝟕
5
18
9
15
10
The centroid is the point of
concurrency of the medians
of a triangle, therefor…
2 2x − 3 = x + 18
4x − 6 = x + 18
3x − 6 = 18
3x = 24
x=8
Find the length of the base in the rectangle below.
Use the Pythagorean
Theorem to find the length
of the hypotenuse.
𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐
8
62 + 𝑏 2 = 102
36 + 𝑏 2 = 100
−36
−36
𝑏 2 = 64
𝑏=8
If a 25 foot ladder leans against the side of a building 24 feet
from its base, how high up the building will the ladder reach?
7
Use the Pythagorean
Theorem to find the length
of the hypotenuse.
𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐
242 + 𝑏 2 = 252
576 + 𝑏 2 = 625
−576
−576
𝑏 2 = 49
𝑏=7
In the triangle below, write the sine, cosine, and tangent ratios
for angle B and angle C.
hypotenuse
leg
leg
𝐴𝐶
sin ∠𝐵 =
𝐵𝐶
𝐴𝐵
sin ∠𝐶 =
𝐵𝐶
𝐴𝐵
cos ∠𝐵 =
𝐵𝐶
𝐴𝐶
cos ∠𝐶 =
𝐵𝐶
𝐴𝐶
tan ∠𝐵 =
𝐴𝐵
𝐴𝐵
tan ∠𝐶 =
𝐴𝐶
opposite leg
𝑠𝑖𝑛 =
hypotenuse
adjacent leg
𝑐𝑜𝑠 =
hypotenuse
opposite leg
𝑡𝑎𝑛 =
adjacent leg
𝑨𝑪 ≅ 𝑨𝑪 𝒃𝒚 𝒕𝒉𝒆 𝒓𝒆𝒇𝒍𝒆𝒙𝒊𝒗𝒆 𝒑𝒓𝒐𝒑𝒆𝒓𝒕𝒚
𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆
∆𝑩𝑨𝑪 ≅ ∆𝑫𝑨𝑪 𝒃𝒚 𝑯𝑳 (𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 − 𝒍𝒆𝒈)
𝒍𝒆𝒈
𝑳
𝑯
𝑆𝐷 ⊥ 𝐻𝑇 𝑎𝑛𝑑 𝑆𝐻 ≅ 𝑆𝑇
𝐺𝑖𝑣𝑒𝑛
∠𝑆𝐷𝐻 𝑎𝑛𝑑 ∠𝑆𝐷𝑇 𝑎𝑟𝑒 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒𝑠
𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑙𝑖𝑛𝑒𝑠
∆𝑆𝐷𝐻 and ∆𝑆𝐷𝑇 𝑎𝑟𝑒 𝑟𝑖𝑔ℎ𝑡 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠
𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑜𝑛 𝑜𝑓 𝑟𝑖𝑔ℎ𝑡 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠
𝑆𝐷 ≅ 𝑆𝐷
𝑅𝑒𝑓𝑙𝑒𝑥𝑖𝑣𝑒 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦
∆𝑆𝐻𝐷 ≅ ∆𝑆𝑇𝐷
𝐻𝐿
Definition of right triangles
Converse of Isosceles Triangle Theorem
Reflexive Property
HL
What are the lengths of the missing sides in the triangle?
leg
leg
𝟕
𝟕 𝟐
hypotenuse
In a 𝟒𝟓° − 𝟒𝟓° − 𝟗𝟎°
triangle, the legs are
congruent.
In a 𝟒𝟓° − 𝟒𝟓° − 𝟗𝟎°
triangle, the
hypotenuse is the
length of a leg times
the 𝟐.
𝟕 𝟐
𝟕
Quilt squares are cut on the diagonal to form triangular quilt
pieces. The hypotenuse of the resulting triangles is 10
inches long. What is the side length of each piece? Leave
your answer in simplified radical form.
10 𝑖𝑛.
∟
𝟒𝟓°
𝟒𝟓°
In a 𝟒𝟓° − 𝟒𝟓° − 𝟗𝟎° triangle,
the hypotenuse is the length
of a leg times the 𝟐.
So, the length of a leg is
the length of the
hypotenuse divided by 𝟐.
10
Rationalize the denominator
2
10 2
=5 2
×
=
2
2
2
A conveyor belt carries supplies from the first floor to the
second floor, which is 24 feet higher. The belt makes a 60°
angle with the ground. How far do the supplies travel from one
end of the conveyor belt to the other? Round your answer to
the nearest foot.
opposite leg
𝑠𝑖𝑛 =
hypotenuse
24 𝑓𝑡.
∟
60°
24
≈ 27.7 𝑓𝑡.
sin 60° =
𝑥
Use a scientific calculator and divide
24 by the sin of 60 degrees…
In the triangle below, find the values of 𝑎 and 𝑏.
The altitude (a) of the
right triangle is the
geometric mean of the
two parts of the
hypotenuse.
𝑎
2
=
32 𝑎
The leg (b) of the right
triangle is the geometric
mean of the entire
hypotenuse and the
adjacent part of the
hypotenuse..
Take the
square root of
each side…
𝑎2 = 64
Because distance
cannot be negative
𝑎 = ±8
𝑏
2
𝑎=8
=
34 𝑏
Cross Multiply
𝑏 2 = 68
𝑏 = ± 68
Cross Multiply
Take the
square root of
each side…
Jason wants to walk the shortest distance to get from the
parking lot to the beach.
Use the Pythagorean
Theorem to find the length
of the hypotenuse.
50 𝑚
32 𝑚
The altitude (a) of the𝟐
𝟐 = 𝒄𝟐
𝒂
+
𝒃
right triangle is the
geometric mean of the2
2 = 𝑐2
30
+
40
two parts of the
hypotenuse.
2
900 + 1600 = 𝑐
𝑎
32
=
18
𝑎
2500 = 𝑐 2
50 = 𝑐
𝑎2 = 576
𝑎 = 24 𝑚
a. How far is the spot on the beach from the parking lot?
b. How far will his place on the beach be from the refreshment stand?
50 − 18 = 32 𝑚
Kristen lives directly east of the park. The football field is directly south of the
park. The library sits on the line formed between Kristen’s home and the
football field at the exact point where an altitude to the right triangle formed by
her home, the park, and the football field could be drawn. The library is 2 miles
from her home. The football field is 5 miles from the library
The leg (b) of the right
triangle is the geometric
mean of the entire
hypotenuse and the
adjacent part of the
hypotenuse..
10 𝑚
35 𝑚
𝑏 7
=
5 𝑏
𝑏 2 = 35
𝑏 = 35 𝑚
a.
b.
How far is library from the park?
How far is the park from the football field?
The altitude (a) of the
right triangle is the
geometric mean of the
two parts of the
hypotenuse.
𝑎 2
=
5 𝑎
𝑎2 = 10
𝑎 = 10 𝑚
Find the length of 𝐴𝐵.
𝐨𝐩𝐩𝐨𝐬𝐢𝐭𝐞 𝐥𝐞𝐠
𝐒𝐢𝐧𝐞 =
𝐡𝐲𝐩𝐨𝐭𝐞𝐧𝐮𝐬𝐞
𝒙
𝟒. 𝟎𝟗
𝒙
𝑺𝒊𝒏 𝟏𝟕 =
𝟏𝟒
𝑺𝒊𝒏 𝟏𝟕 𝟏𝟒 = 𝟒. 𝟎𝟗
Use a scientific
calculator and multiply
the Sine of 17 degrees
by 14…
A large totem pole in the state of Washington is 100 feet tall.
At a particular time of day, the totem pole casts a 249-footlong shadow. Find the measure of ∠to the nearest degree.
𝟐𝟐°
100
tan 𝑥° =
≈ 2.188°
249
Use a scientific calculator
𝒕𝒂𝒏−𝟏 (𝟏𝟎𝟎 ÷ 𝟐𝟒𝟗)
opposite leg
𝑡𝑎𝑛 =
adjacent leg