Download STAR CITY Math / Geometry / Midsegments Name Teacher Period

STAR CITY
Math / Geometry / Midsegments
Name _________________________________
Teacher ____________________ Period _____
2.
Use the figure below to answer question 1.
1.
A triangular jogging path ABE is located in the
park. The dimensions are EB = 92 ft.,
AE = 78 ft., and AB = 140 ft. The park
administrator wants to make an additional path
¯ . What is the
using the midsegment DC
perimeter of the new path ABCD?
¯?
What is the length of OR
A.
155 ft.
A.
2.5 ft.
B.
295 ft.
B.
3 ft.
C.
310 ft.
C.
3.5 ft.
D.
380 ft.
D.
10 ft.
1
STAR CITY – Math – Geometry – Midsegments
Use the figure below to answer question 3.
3.
Use the figure below to answer question 4.
In ΔHNS , A and D are midpoints. What is the
value of x?
A.
8
4.
B.
10
The figure above represents a light pole on a
¯ ⊥ GT
¯ and H is the midpoint of
runway. If LH
¯ , which term does not identify IH̄ ?
TG
C.
11
A.
midsegment
D.
15
B.
median
C.
altitude
D.
perpendicular bisector
2
STAR CITY – Math – Geometry – Midsegments
5.
¯ as a midsegment?
Which triangle has BD
Use the figure below to answer question 6.
A.
B.
6.
C.
D.
3
¯?
What is the length of the midsegment BE
A.
4 cm
B.
8 cm
C.
11 cm
D.
16 cm
STAR CITY
Open-Response / Geometry / Midsegments
Name _________________________________
Teacher ____________________ Period _____
GEOMETRY OPEN-RESPONSE ITEM A
A.
¯ and AF
¯ , that intersect to form an angle of 52°. The
An A-frame bookshelf has two congruent supports, AE
shelves divide each support into four congruent lengths. The distance between each pair of shelves is 18 ½
inches.
1. What is the measure of ∠ DCH ? Show or explain your work.
2. If the distance between supports for the middle shelf is 32 inches, find the distance between the
¯ . Show or
¯ and the distance between the supports on the floor, EF
supports for the top shelf, BJ
explain your work.
¯ rounded to the nearest hundredth of an inch? Show or explain your work.
3. What is the length of HG
BE SURE TO LABEL YOUR RESPONSE (1), (2), AND (3).
4
STAR CITY
Student Answer Document / Geometry / Midsegments
Name _________________________________
Teacher ____________________ Period _____
A
5
GEOMETRY OPEN-RESPONSE ITEM A (T.2.G.3E, T.2.G.1B, T.2.G.1C)
A.
¯ and AF
¯ , that intersect to form an angle of 52°. The shelves divide each
An A-frame bookshelf has two congruent supports, AE
support into four congruent lengths. The distance between each pair of shelves is 18 ½ inches.
1. What is the measure of ∠ DCH ? Show or explain your work.
2. If the distance between supports for the middle shelf is 32 inches, find the distance between the supports for the top shelf,
¯ and the distance between the supports on the floor, EF
¯ . Show or explain your work.
BJ
¯ rounded to the nearest hundredth of an inch? Show or explain your work.
3. What is the length of HG
BE SURE TO LABEL YOUR RESPONSE (1), (2), AND (3).
RUBRIC FOR GEOMETRY OPEN-RESPONSE ITEM A (T.2.G.3E, T.2.G.1B, T.2.G.1C)
Description
Score
4
Student scores 4 points. All explanations are clear and correct.
3
Student scores 3 - 3 ½ points.
2
Student scores 2 - 2 ½ points.
1
Student scores ½ - 1 ½ points.
or
The response demonstrates minimal understanding of solving problems involving
relationships in trapezoids and triangles.
0
Student shows no understanding of the problem.
B
Solution and Scoring
Scoring:
Part 1: Score 1 point for finding the correct measure of ∠ DCH with work shown or explanation given.
OR
Score ½ point for finding the correct measure of ∠ DCH with no work shown or explanation given.
OR
Score ½ point for correct procedure to find the measure of ∠ DCH with calculation error in work
shown.
Part 2: Score 2 points total:
¯ with work shown or explanation given.
Score 1 point finding the correct length of BJ
OR
¯ with no work shown or explanation given.
Score ½ point for finding the correct measure of BJ
OR
¯ with calculation error in work shown.
Score ½ point for correct procedure to find the measure of BJ
AND
¯ with work shown or explanation given.
Score 1 point finding the correct length of EF
OR
¯ with no work shown or explanation given.
Score ½ point for finding the correct measure of EF
OR
¯ with calculation error in work shown.
Score ½ point for correct procedure to find the measure of EF
¯ with work shown or explanation given.
Part 3: Score 1 point finding the correct length of HG
OR
¯ with no work shown or explanation given.
Score ½ point for finding the correct measure of HG
OR
¯ with calculation error in work shown.
Score ½ point for correct procedure to find the measure of HG
Note:
In Part 2 and Part 3, including units is a 3 / 4 issue only. To receive a score of 4, the student must
include correct units, but there is no deduction for missing or incorrect units below a score of 4.
Sample Solution 1:
Part 1:
1
2
(180o − 52o) = 64o
1
2
∠ ABJ = 64o = ∠ CDG
∠ AJB = 64o = ∠ HGD
∠ ABJ = 64o
OR
corresponding angles are congruent
1
o
(
2 360
∠ BCH + ∠ DCH = 180olinear pairs
− 2(64o)) = 116o
ΔABJ is similar to ΔACH by
∠ ABJ = ∠ BCH
corresponding angles are congruent
∠ DCH = 180o − 64o = 116o
∠ DCH = 116o
Part 2:
(180o − 52o) = 64o
OR
¯ is the midsegment of ΔACH .
BJ
AA similarity
¯=
BJ
1
(
)
2 CH
¯=
BJ
1
( )
2 32
¯ = 16 inches
BJ
ΔAEF is 4 times greater than ΔABJ .
¯ ) = 4(16) = 64 inches.
4( BJ
Part 3:
( AJ
¯ )2 =
2
1 2
¯ ) + (18 2 )
( 12 BJ
(
1
( AJ
¯ )2 = (8)2 + 18 2
)2
( AJ
¯)2 = 406.25
¯ = 20.16inches
AJ
¯ = HG
¯
AJ
¯ = 20.16inches
HG
Sample Solution 2:
Part 1:
1
2
(180o − 52o) = 64o
1
2
∠ ABJ = 64o = ∠ CDG
∠ ABJ = 64o
∠ AJB = 64o = ∠ HGD
OR
corresponding angles are congruent
1
o
(
2 360
(180o − 52o) = 64o
∠ ABJ = ∠ BCH
corresponding angles are congruent
∠ BCH + ∠ DCH = 180olinear pairs
− 2(64o)) = 116o
∠ DCH = 180o − 64o = 116o
∠ DCH = 116o
Part 2: Using proportions with Triangle ABJ and Triangle
congruent from Part 1),
ACH (because we know the angles are
1
Height ofΔABJ
BJ (Base ofΔABJ )
=
18 2
Height ofΔACH
,
CH (Base ofΔACH )
x
¯ as the median of trapezoid
Using CH
BJGD,
=
37
32
¯ = 16 in.
x = 16 in. so BJ
¯ as the median of trapezoid
Now use DG
CHFE, then
¯=
CH
32 =
Part 3: ( AJ
¯ )2 =
1
(¯
2 BJ
1
(
2 16
¯)
+ DG
¯)
+ DG
¯=
DG
48 =
1
(¯
2 CH
1
(
2 32
¯)
+ EF
¯)
+ EF
¯)
32 × 2 = (16 + DG
¯)
48 × 2 = (32 + EF
¯)
64 = (16 + DG
¯)
96 = (32 + EF
¯
64 − 16 = DG
¯
96 − 32 = EF
¯ = 48in.
DG
¯ = 64in.
EF
2
1 2
¯ ) + (18 2 )
( 12 BJ
(
1
( AJ
¯ )2 = (8)2 + 18 2
( AJ
¯)2 = 406.25
¯ = 20.16inches
AJ
¯ = HG
¯
AJ
¯ = 20.16inches
HG
)2
STAR CITY
Math / Geometry / Midsegments
Answer Key and Alignment
Answer
1. A
2. B
3. C
4. A
5. B
6. A
Source
TLI
TLI
TLI
TLI
TLI
TLI
SLE
T.2.G.3e
T.2.G.3e
T.2.G.3e
T.2.G.3e
T.2.G.3e
T.2.G.3e