G.CO.12 ASSESSMENT – PATTERSON 1 MULTIPLE CHOICE 1. What is the best description for the distance from Point A to Point B? A
B
A) AB B) AB C) about 2 cm C
2. What is the best description for the distance from Point A to Point B? E
F
B
A) CD + 2EF B) CD -­‐ EF C) 2CD -­‐ EF A
A) CD -­‐ EF B) 2CD -­‐ 2EF D) 2CD + EF C
D
A
3. What is the best description for the distance from Point A to Point B? D) about 1.5 inches D
E
F
B
C) 3CD -­‐ 2EF 4. A teacher finds a paper on the ground in the classroom. When she looks at it carefully she realizes it is from her geometry class because it has a construction on it. Which of the following constructions is NOT FOUND directly from this student’s work? A) The midpoint of AB B) The perpendicular bisector of AB C) A perpendicular line to AB D) The angle bisector of ∠CAB 5. Which construction is represented by these construction marks? A) Copying ∠ABC B) The perpendicular bisector of BC C) The angle bisector of ∠ABC D) A perpendicular line AC D) 3EF – 2CD C
A
B
D
A
B
C
6. When doing a construction, which geometric instrument is used to measure the length of a segment? A) A ruler B) A compass C) A protractor 3. D 6. B D) A straightedge Answers: 1. B 2. C 4. D 5. C G.CO.12 ASSESSMENT – PATTERSON 2 SHORT ANSWER (Explain) 1. What is congruent in this diagram? Why are they congruent? A
D
C
E
B
Answer is BD ≅ BE (same radii) or ∠ABC ≅ ∠DBE (same angle) 2. Jeff is constructing the angle bisector of ∠DBE. What is the next step? Be very specific as to what he should do next. D
E
B
Answer is “Keep the measure of the compass the same as what you just used to create the arc in the interior of the angle. Place the pointer of the compass on Point D and create a new arc (using that same measure) until it intersects the given arc. The intersection of these two arcs is a point on the angle bisector.” 3. Nancy is copying ∠ABC. What is the next step? Be very specific as to what she should do next. A
D
C
B
B'
L'
L
Answer is “Place your compass pointer on Point L and stretch it until it reaches Point D. You now have the measure LD. Place your pointer now on Point L’ and create an arc that intersects the one you just completed. Where these two intersect is the Point D’.” 4. When you do a midpoint construction of CD , you must stretch your compass so that it is greater than half the distance of CD . Why do you have to do this? Why couldn’t you use a distance smaller than half of CD ? If the distance is smaller than half the distance then they will not intersect. They must intersection to locate two points that are both equidistant from Point C and Point D. G.CO.12 ASSESSMENT – PATTERSON 3 5. Lindsay notices that while doing a construction a ‘hidden’ shape appeared – a rhombus. Where is the rhombus hidden in this shape? Draw in the segments that form the rhombus and explain why must it be a rhombus? C
A
B
D
The rhombus is made with points A, C, B & D. The reason it is a rhombus because the measurement AC was used to create AD, BD & BC. Because the same radii were used for all points, the 4 lengths are equal. Rhombus is defined by 4 equal sides. 5. An Isosceles triangle is a triangle that has at least two equal sides. When doing the perpendicular bisector construction Jennifer spots a hidden isosceles triangle. Can you find a hidden isosceles triangle in the construction? Diagram it and then explain why you think it is isosceles? C
A
B
D
There are lots of hidden isosceles triangles found in this diagram. Most students will list ΔACB or ΔADB. The reason they are isosceles is that AB = BC and AD = BD. They are equal because we keep the measure the same. ΔDAC is also isosceles because AC = AD. ΔBCD is also isosceles because BC = BD. 6. Victoria is copying ∠ABC. What is the next step? Be very specific as to what she should do next. A
B
C
B'
C'
The next step is to measure the distance CA, and then placing the pointer on the Point C and making an arc that intersects the previously created one. This intersection is A’. Now you can form ∠A’B’C’. 7. A teacher instructs the class to construct one-­‐fourth the length of a segment. Michael pulls out his ruler and measures the segment to the nearest millimeter and then divides the length by four. He marks this distance from one of the endpoints. Has he done this correctly? Explain. No! We do not use a ruler to measure or construct when we are doing a construction. A ruler is a straightedge – to draw straight lines – Not Measure!! We would do two midpoint point constructions to divide the line into four congruent pieces. G.CO.12 ASSESSMENT – PATTERSON 4 8. What is the difference between ∠ABC and m∠ABC? ∠ABC is referring to the angle -­‐ the object. m∠ABC is referring to the measurement of the angle. 9. A student is told that AB and CD have equal lengths. The student writes down AB = CD . What is wrong with this mathematical statement? The notation is about objects and when we speak of objects we use congruence AB ≅ CD . But to correctly write that the lengths of these two segments are equal it would need to be notated AB = CD . 10. Use the diagram to complete the relationship. (In diagrams 1, 2 and 4 the compass was constant for each individual construction.) C
C
F
E
E'
C
E
D
A
D
D
F'
D
G
F
a) DF = ________ E
F
F
B
E
C
b) EF ≅ _______ c) CE = ________ e) ________= ________
g) m∠ABF = ________
d) ED ≅ ________ f) ________≅ ________ h) BE ≅ ________ Answers a) DF = EF = DC = EC c) ED e) EF = E’F’ g) m∠FBC b) DF , DC or EC d) CE f) EF ≅ E ' F ' h) BD, EF , or DF G.CO.12 ASSESSMENT – PATTERSON 5 LONG ANSWER (Constructions) 1
1. Given the two segments, using a compass and straightedge, construct the exact length of 2 AB − CD . 2
A
B C
D
Copy AB twice, and then subtract back one-­‐half of CD. To get one-­‐half of CD, the student has to do a midpoint construction on CD . A
B
C
D
2. Given ∠ABC, create an angle exactly twice the size of ∠ABC. A
C
B
The student copies the angle, and then copies it again but the second one share one ray with the first copy. A
F
H
C
I
B
G
suur
3. Given the Point C and AB , determine all points that are a distance of suur
1
AB away from Point C and on AB . 2
G.CO.12 ASSESSMENT – PATTERSON 6 C
A
B
The student should first do the midpoint construction of AB , and then use that as a radius with center C. suur
That circle will intersect twice with AB . These are your two solutions. C
A
D
B
E
uuur
uuur
4. Copy ∠ABC so that B’ is on BC and Point A’ is on the same side of BC as Point A. A
C
B
B'
Create C’ by measuring BC and copying it onto the ray. Measure BA with your compass and while having the pointer of the compass at B’ creating an arc on the same side of the ray as Point A. Measure CA with your compass and while having the pointer of the compass at C’ creating an arc on the same side of the ray as Point A. These two arcs will intersect giving us Point A’ completing the copy of the angle. A'
A
B
C'
C
B'
G.CO.12 ASSESSMENT – PATTERSON 7 5. Construct the angle bisector of ∠ABC. A
B
C
Create any arc that intersects both rays of the angle. Name those two new intersections D and E. Using the same measurement, place your pointer at Point D and create an arc in the interior of the angle. Repeat that same step but with your pointer at Point E. The intersection of those two arcs is point F, a point on the angle bisector of ∠ABC. A
D
F
B
E
C
6. Create a perpendicular line to AB through Point C. C
A
B
Placing the pointer of your compass on Point C, extend its length such that it intersects AB twice. Name those points D and E. Keeping the measure of your compass the same, place the pointer of the compass at D and create an arc on the side of AB that Point C is not on. Do the same for Point E. These two arcs will intersect to form Point F. Using your straightedge draw CF . G.CO.12 ASSESSMENT – PATTERSON 8 C
B
A D
E
F
7. Construct 1
AB . 4
A
B
Perform the midpoint construction to get Point E. Then perform the midpoint construction again using 1
segment AE to get Point H. AH = AB . 4
C
F
A
H
E
B
G
D
uuur
8. Construct a parallel line to BC through Point A. G.CO.12 ASSESSMENT – PATTERSON 9 A
C
B
uuur
Place a Point B’ somewhere on BC . Place your compass pointer on Point B and stretch the compass to uuur
measure BA. Create an arc so that it intersects BC . Name this point L. Now copy ∠ABL to ∠A’B’L’. Using a straightedge draw the line through A and A’. This is a parallel line. A'
A
C
B'
B
L'
L