Transcript OPEN RESPONSE QUESTION FROM MCAS SPRING 2004, …

INTRODUCTION
OPEN RESPONSE QUESTION
FROM MCAS FALL RETEST
2003, GRADE 10, #17
The following is an “Open Response” question for you to
practice answering these types of questions in preparation
for the State Mandated MCAS. This exercise will help to
walk you through the process. There are also some
reference links to give you some extra help.
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HOW TO ANSWER AN OPEN
RESPONSE QUESTION
Be sure to
 Read all parts of each question carefully.
 Make each response as clear, complete
and accurate as you can.
 Check all your work!
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TOPIC
Apply similarity correspondences (ex. ΔABC ~
ΔXYZ) and properties of the figures to find missing
parts of geometric figures, and provide logical
justification.
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QUESTION #17
To measure the width of a stream indirectly, Claude placed
four stakes in the ground at points B, C, D, and E. He used a
rock on the opposite bank to determine point A. Triangles
ABC and ADE are formed, as shown in the diagram below.
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QUESTION #17
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Part A. Explain how you can show BCA is
congruent to  DEA.
Hint: Separate the diagram into two separate triangles.
Additional
Hint
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Part B. Explain how you know BCA is
similar to DEA.
Hint
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Part C. Write a proportion or an equation that
can be used to determine the distance
(indicated by d in the diagram) across the
stream.
Hint
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Part D. What is the distance across the
stream? Show or explain how you obtained
your answer.
Hint
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REFERENCES
If you are having any trouble with this problem there are some
links below that will help you with the standards involved with
this question:
http://mathforum.org/dr.math/
http://www.algebra.com/algebra/homework/coordinate/
http://school.discovery.com/homeworkhelp/webmath/
http://www.doe.mass.edu/mcas/
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EVALUATION
This is the rubric that the state uses to rate responses to question 17. How did you do?
4
The student response demonstrates an exemplary understanding of the Geometry concepts involved in
working with similar triangles calculating lengths of sides through proportions.
3
The student response demonstrates a good understanding of the Geometry concepts involved in working
with similar triangles calculating lengths of sides through proportions. Although there is significant
evidence that the student was able to recognize and apply the concepts involved, some aspect of the response
is flawed. As a result the response merits 3 points.
2
The student response contains fair evidence of an understanding of the Geometry concepts involved in
working with similar triangles calculating lengths of sides through proportions. While some aspects of the
task are completed correctly, others are not. The mixed evidence provided by the student merits 2 points.
1
The student response contains only minimal evidence of an understanding of the Geometry concepts
involved in working with similar triangles calculating lengths of sides through proportions.
0
The student response contains insufficient evidence of an understanding of the Geometry concepts involved
in working with similar triangles calculating lengths of sides through proportions.
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SOLUTIONS
Solution for A
Solution for B
Solution for C
Solution for D
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CONCLUSION
Hopefully, this has been a helpful experience for you.
The only way to improve is to practice, and there are
other problems like this one available for you to try if
you are up to the challenge!
-- or --
 Call a friend, have them do the same problem
and you can talk about it.
-- or - Take a break, you’ve earned it!
to try another Problem 
Additional Hint for Part A
A
d
A
d + 6 ft
B 28 ft C
D
32 ft
E
When you separate the two triangles, notice that  B and  D
are congruent and that  A is the same angle in each of the two
triangles. If you know that two angles of a triangle are congruent
to two angles of another triangle, what do you know about the
third angle?
Go Back
Hint for Part B
Two triangles are similar if all corresponding angles
are congruent and all corresponding sides are in
proportion.
In order to prove triangles are similar you must
prove one of the following similarities:
SAS
Similarity
SSS
Similarity
Back to Part B
AA
Similarity
SAS Similarity
SAS Similarity – Prove that two corresponding
sides are in proportion and that the
corresponding included angles, the angles
located in between the sides, are congruent.
Ex.
Since
10
20
A
7
= 14
10 ft
and B = D = 90 º
F
20 ft
B 7 ft C
then ABC ~ FDE by SAS
Similarity.
Back to Hint Page
D
14 ft
E
SSS Similarity
a. SSS Similarity – Prove that all corresponding
sides are in proportion.
Ex.
Since
8
20
F
A
8 ft
10
= 25
6
= 15
10 ft
20 ft
25 ft
B 6 ft C
then ABC ~ FDE by SSS
similarity.
Back to Hint Page
D
15 ft
E
AA Similarity
AA Similarity – Prove that two corresponding
angles are congruent.
A
F
Ex.
Since
A = F = 65º
B = D = 90º
B
65º
then  ABC ~  FDE by AA
Similarity.
Back to Hint Page
C
D
65º
E
Hint for Part C
Once you know the
triangles are similar,
you can set up a
proportion using the
sides lengths that are
given. In this problem,
use
AB
BC
AD = DE
A
B
Back to Part C
A
C
D
E
Hint for Part D
Solve the proportion you wrote in part C by cross
multiplying:
AB
BC
d
28
means
that
AD = DE
d+6 = 32
When you get your solution for d, you will have
found the distance across the lake.
Back to Part D
Solution A
 ABC and  DEA are congruent because they are
both right angles. Angle  BAC and  DAC are
congruent because they are the same angle,
reflexive property. There is a theorem that states
if two angles of one triangle are congruent to
two angles of another triangle, then the third
angle is congruent.
As an alternate solution, you could use the sum of
the triangles is 180 to explain why the two angles
are congruent.
Back to Solution Page
Solution B
Since from part A you know that two angles
of one triangle are congruent to two
angles of another triangle, then ABE ~
ADE by the AA Similarity Postulate.
Back to Solution Page
Solution C
Since you know that:
AB
BC
AD = DE
Plug in the values and write the proportion:
d
28
d+6 = 32
Back to Solution Page
From part C:
d
28
d+6 = 32
Solution D
After cross-multiplying:
32d = 28(d+6)
32d = 28d + 168
4d = 168
d = 42
Back to Solution Page
I’m done!
So the distance across the stream, AB, is 42 feet.