# Algebra questions

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College Geometry — MATH 3331 Fall 2017 Take-home Final Exam This is an exam, and you are expected to work on it individually. You must show all work, and provide explanations for the steps you are taking. Proofs must be formal, with the word “Theorem,” a statement of the theorem you are proving, the word “Proof,” a complete proof with statements and reasons, and an end-ofproof marker. In proofs you may refer to theorems in the text, as well as postulates and definitions. Make sure to include the number of the theorem, postulate, or definition to which you are referring. This figure is used for the first two problems of the exam. Let ABCD be a square. Bisect AD at E. Connect EB and extend EA to F so that EF ∼ = EB. Construct the square AF GH with point H on AB. F A G H B E D C 1. If the length of the side of the square ABCD is 1, find the lengths of EB and AH. Be sure to include all the details on how you got your answer. 2. Prove that HB HA = . HA AB Do not use any specific numerical lengths for any line segments, this is to be done in general. 3. In ancient Egypt, surveyors were called “rope-stretchers”. This is because one of the main surveying tools was a long loop of rope with twelve equally spaces knots tied into it, like a clock. 11 12 1 10 2 9 3 8 4 7 6 5 Three surveyors would grab the knots at “3 o’clock”, “7 o’clock”, and “12 o’clock”, and pull the rope taught. What kind of triangle is formed by doing this? What kind of angle is formed at the “7 o’clock” point? Of course, you must prove your answers. Theorem (Theorem of Menelaus). Let 4ABC be a triangle. Let points L, M , and N be ←→ ←→ ←→ points on the lines BC, AC, and AB (Note that these points are on the lines, not the line segments.) C M L A B N Then the points L, M , and N are collinear if and only if AN BL CM · · = −1. N B LC M A 4. Prove the “only if” part of the theorem. That is, assume that the points are collinear and prove the equation true. You can assume that two of the points lie on the triangle, as L and M do in the figure. Drop perpendiculars from A, B, and C onto the line ←−−−−−−→ M − L − N . Use similar triangles to express each of the fractions in the equation in terms of the lengths of the perpendiculars; then finish with some algebra. 5. Prove that “if” part of the theorem. That is, assume that the equation is true, and then prove that point N must be on line LM . You can assume that LM is not parallel to AB, and so LM intersects AB at a point N 0 . Use the “only if” part of the theorem and some algebra to prove that N and N 0 are the same point.

Prof.Demidko
School: UIUC

Attached.

Surname 1

1. Given vector quantities below;
⃗ = (2*106î -3*106k̂) m/s= (VXî -VZk̂)
𝑉
⃗ = (0.03î -0.15ĵ ) T= (BXî -BYĵ )
𝐵
a. determine the magnetic force 𝐹 B
⃗ *𝐵
⃗)
𝐹 B= -q (𝑉
⃗ *𝐵
⃗ = (VXî -VZk̂)*(BXî -BYĵ ) = (VXBX) (i*i)-(VXBY) (i.j)-(VZBX) (k.i) + (VZBY) (k.j)
𝑉
𝐹 B=-q [-(VXBYi+VZBXj- VZBYk)]
𝐹 B=-q [-(-2*106*0.15) k̂ +3*106*0.03ĵ +3*106*0.15 î ] simplifying,
𝐹 B=-q (0.3 k̂ +0.09ĵ +0.45î )*106N
𝐹 B=1.602*10-19(0.3 k̂ +0.09ĵ +0.45î )*106N
𝐹 B= (4.8+1.4+7.2)*10-14J
b. 𝐹 B=q𝐸⃗ ;making𝐸⃗ the subject of the formula
𝐸⃗ =𝐹 B/q
1
𝐸⃗ =-𝑞(0.3k̂+0.09ĵ +0.45î )*106N but q= (1.602*10-19C)

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Anonymous
Outstanding Job!!!!

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