Practice Problems
Brice Chen's Practice Problems
#1- A straight sidewalk leads from Brice's house, A, to the school, C. Brice has a friend living in a house, B, on the sidewalk, which, coincidentally, is halfway between Brice's house and the school. If the distance between Brice's house and the school is 98 meters, the distance between Brice's house and his friend's house is 9x+2y, and the distance between his friend's house and the school is 12x-y-9, what is the distance in meters of x?
Solving- First off, the problem is telling you that the sidewalk between Brice's house, A, and the school, c, is a line segment, since it refers to it as a straight line between two points. Also, B, Brice's friend's house is the midpoint of the segment, since it is halfway in between A and C. That means segment AB equals segment BC. Then we substitute in the parts, so segment AB=segment BC is 9x+2y=12x-y-9. Also, segment AB+segment BC equals segment AC. The numbers for those are 9x+2y+12x-y-9=98, simplified into 21x+y=107. In addition, both segment AB and BC are equal to 49, since they are congruent. 9x+2y=12x-y-9 breaks down into -3x+3y=-9, which can be broken down further into x=y+3, by dividing the entire thing by -3 and moving the y to the other side. This is then substituted in 9x+2y=49, turning it into 9(y+3)+2y=49. After distributing and simplifying, the equation turns into 11y+27=49. You then subtract 27 from both sides and get 11y=22, and divided that turns into y=2. After plugging it back in the same equation, you have 9x+4=49. Subtract 4 from both sides and it leaves 9x=45. Divide it by 9 and X=5 which is the answer.
In numbers-
Step 1
AC=98, AB congruent with BC Given
AB and BC= 49 Definition of midpoint
AB=BC Definition of congruence
9x+2y=12x-y-9 Substitution
-3x+2y=-y-9 SPOE
-3x+3y=-9 APOE
x-y=3 DPOE
x=y+3 APOE
Step 2
9x+2y=49 Given
9(y+3)+2y=49 Substitution
9y+27+2y=49 Distribution
11y+27=49 Simplify
11y=22 SPOE
y=2 DPOE
Step 3
9x+2y=49 Given
9x+4=49 Substitution
9x=45 SPOE
x=5 DPOE
#2- There is a intersection of two straight roads. One road has the points ABC, and the other road has the points DBF. A gas station opened on angle ABF of the intersection, and wants to know, how many degrees is angle ABF. If angle DBC is 9x+18 degrees, and angle ABD is 5x-6 degrees, what is the measure of angle ABF, in degrees?
Solving- Angle DBC is equal to angle ABF, since they are vertical angles. Angle ABD and angle DBC are a linear pair, and the sum of them is 180 degrees. Angle ABD+angle DBC=180. 9x+18+5x-6=180, from substituting in the values. 14x+12=180 comes from simplifying. Subtract 12 from both sides and get 14x= 168. Divide that and x=12. Put that in the equation, but make it angle DBC=180-angle ABD. Then, substitute in and get 9(12)+18=180-5(12)-(-6). Distribute for 108+18=180-60+6 Simplify for 126=126, so angle DBC=126, and angle ABF=126 degrees, the answer.
In numbers-
Step 1
DBC=ABF, ABD+DBC=180, DBC=9x+18, ABD=5x-6 Given
9x+18+5x-6=180 Substitution
14x+12=180 Simplify
14x=168 SPOE
x=12 DPOE
Step 2
DBC=180-ABD Given
9(12)+18=180-5(12)-(-6) Substitution
108+18=180-60+6 Distribute
126=126 Simplify
ABF=126 Substitution
Solving- First off, the problem is telling you that the sidewalk between Brice's house, A, and the school, c, is a line segment, since it refers to it as a straight line between two points. Also, B, Brice's friend's house is the midpoint of the segment, since it is halfway in between A and C. That means segment AB equals segment BC. Then we substitute in the parts, so segment AB=segment BC is 9x+2y=12x-y-9. Also, segment AB+segment BC equals segment AC. The numbers for those are 9x+2y+12x-y-9=98, simplified into 21x+y=107. In addition, both segment AB and BC are equal to 49, since they are congruent. 9x+2y=12x-y-9 breaks down into -3x+3y=-9, which can be broken down further into x=y+3, by dividing the entire thing by -3 and moving the y to the other side. This is then substituted in 9x+2y=49, turning it into 9(y+3)+2y=49. After distributing and simplifying, the equation turns into 11y+27=49. You then subtract 27 from both sides and get 11y=22, and divided that turns into y=2. After plugging it back in the same equation, you have 9x+4=49. Subtract 4 from both sides and it leaves 9x=45. Divide it by 9 and X=5 which is the answer.
In numbers-
Step 1
AC=98, AB congruent with BC Given
AB and BC= 49 Definition of midpoint
AB=BC Definition of congruence
9x+2y=12x-y-9 Substitution
-3x+2y=-y-9 SPOE
-3x+3y=-9 APOE
x-y=3 DPOE
x=y+3 APOE
Step 2
9x+2y=49 Given
9(y+3)+2y=49 Substitution
9y+27+2y=49 Distribution
11y+27=49 Simplify
11y=22 SPOE
y=2 DPOE
Step 3
9x+2y=49 Given
9x+4=49 Substitution
9x=45 SPOE
x=5 DPOE
#2- There is a intersection of two straight roads. One road has the points ABC, and the other road has the points DBF. A gas station opened on angle ABF of the intersection, and wants to know, how many degrees is angle ABF. If angle DBC is 9x+18 degrees, and angle ABD is 5x-6 degrees, what is the measure of angle ABF, in degrees?
Solving- Angle DBC is equal to angle ABF, since they are vertical angles. Angle ABD and angle DBC are a linear pair, and the sum of them is 180 degrees. Angle ABD+angle DBC=180. 9x+18+5x-6=180, from substituting in the values. 14x+12=180 comes from simplifying. Subtract 12 from both sides and get 14x= 168. Divide that and x=12. Put that in the equation, but make it angle DBC=180-angle ABD. Then, substitute in and get 9(12)+18=180-5(12)-(-6). Distribute for 108+18=180-60+6 Simplify for 126=126, so angle DBC=126, and angle ABF=126 degrees, the answer.
In numbers-
Step 1
DBC=ABF, ABD+DBC=180, DBC=9x+18, ABD=5x-6 Given
9x+18+5x-6=180 Substitution
14x+12=180 Simplify
14x=168 SPOE
x=12 DPOE
Step 2
DBC=180-ABD Given
9(12)+18=180-5(12)-(-6) Substitution
108+18=180-60+6 Distribute
126=126 Simplify
ABF=126 Substitution
Vinny Shirvaikar's Practice Problems
1. Aniket eats part of a circular pie. Brice eats twice as much as him, and Dhruv eats three times as much pie as Brice. Finally, Vinny eats 3.5 times as much of the pie as Dhruv. Together, they ate the entire pie. How many degrees of the pie did each of these hungry children eat?
Solution:
Aniket ate 12 degrees of the pie.
Brice ate 24 degrees of the pie.
Dhruv ate 72 degrees of the pie.
Vinny ate 252 degrees of the pie.
Explanation: Since the portion of the pie that Aniket ate is not given, we can represent it with the variable x. Therefore, since Brice ate twice as much as him, his portion would be represented with 2x. Dhruv's portion, then, would be represented with 3(2x), or 6x, and Vinny's with 3.5(6x), or 21x. Because the problem states that the entire pie was eaten, we know that 360 degrees were eaten in total (measure of a circle). So, the equation can be set up as x+2x+6x+21x = 360. By simplifying, we come to 30x = 360. The Division Property of Equality helps us solve x to equal 12 degrees. Now, all that is left is answering the actual question through plugging in. Aniket's is the easiest, because his portion is just x, or 12 degrees. Next, Brice's portion would be 2(12), or 24 degrees. The portion that Dhruv ate is equivalent to 6(12), or 72 degrees of the pie. Finally, Vinny ate 21(12) degrees of the pie, which simplifies to 252 degrees. So, Aniket ate 12 degrees of the pie, Brice ate 24 degrees, Dhruv ate 72 degrees, and Vinny ate 252 degrees. We can check it by making sure that all of these numbers add up to 360, which they do. Neat!
2. Angle A is complementary to Angle B, Angle B is supplementary to Angle C, and m<C+m<D = 360 degrees. m<A will be expressed as a. m<B is equal to a^2+a+82. Find m<D.
Solution: m<D is 268 degrees.
Explanation: Using all of the given information, the first thing we should realize is that a+a^2+a+82 = 90. This simplifies to a^2+2a+82 = 90, which then turns into a^2+2a-8 = 0, an easily factorable quadratic. After factoring, we realize that a comes out to be 2 degrees. Since a and b are complementary, b is obviously 88 degrees. Angle C, therefore, equals 92 degrees, because it is supplementary to Angle B. Lastly, we can plug into the equation m<C+m<D = 360 degrees for m<C and get 92 degrees + m<D = 360 degrees. So, the measure of Angle D, after finishing up this last equation, is 268 degrees.
1. Aniket eats part of a circular pie. Brice eats twice as much as him, and Dhruv eats three times as much pie as Brice. Finally, Vinny eats 3.5 times as much of the pie as Dhruv. Together, they ate the entire pie. How many degrees of the pie did each of these hungry children eat?
Solution:
Aniket ate 12 degrees of the pie.
Brice ate 24 degrees of the pie.
Dhruv ate 72 degrees of the pie.
Vinny ate 252 degrees of the pie.
Explanation: Since the portion of the pie that Aniket ate is not given, we can represent it with the variable x. Therefore, since Brice ate twice as much as him, his portion would be represented with 2x. Dhruv's portion, then, would be represented with 3(2x), or 6x, and Vinny's with 3.5(6x), or 21x. Because the problem states that the entire pie was eaten, we know that 360 degrees were eaten in total (measure of a circle). So, the equation can be set up as x+2x+6x+21x = 360. By simplifying, we come to 30x = 360. The Division Property of Equality helps us solve x to equal 12 degrees. Now, all that is left is answering the actual question through plugging in. Aniket's is the easiest, because his portion is just x, or 12 degrees. Next, Brice's portion would be 2(12), or 24 degrees. The portion that Dhruv ate is equivalent to 6(12), or 72 degrees of the pie. Finally, Vinny ate 21(12) degrees of the pie, which simplifies to 252 degrees. So, Aniket ate 12 degrees of the pie, Brice ate 24 degrees, Dhruv ate 72 degrees, and Vinny ate 252 degrees. We can check it by making sure that all of these numbers add up to 360, which they do. Neat!
2. Angle A is complementary to Angle B, Angle B is supplementary to Angle C, and m<C+m<D = 360 degrees. m<A will be expressed as a. m<B is equal to a^2+a+82. Find m<D.
Solution: m<D is 268 degrees.
Explanation: Using all of the given information, the first thing we should realize is that a+a^2+a+82 = 90. This simplifies to a^2+2a+82 = 90, which then turns into a^2+2a-8 = 0, an easily factorable quadratic. After factoring, we realize that a comes out to be 2 degrees. Since a and b are complementary, b is obviously 88 degrees. Angle C, therefore, equals 92 degrees, because it is supplementary to Angle B. Lastly, we can plug into the equation m<C+m<D = 360 degrees for m<C and get 92 degrees + m<D = 360 degrees. So, the measure of Angle D, after finishing up this last equation, is 268 degrees.
Aniket Matharasi's Practice Problems
1) If I am in athletics, then I play basketball.
Using this conditional statement, show the 5 statements and their truth vlue. If it is false, give a counterexample.
Conditional- If I am in Athletics, then I play basketball. This is false as you can be in athletics and play either football or volleyball.
Converse- If I play basketball, then I am in athletics. This statement is also false, since you can play basketball for fun or play outside of school athletics.
Inverse- If I am not in Athletics, then I do not play basketball. This is a false conjecture, since I can still play basketball even though I am not in athletics.
Contrapositive- If I do not play basketball, then I am not in Athletics. This is false since I can be in Athletics even though I don't play basketball as I would be able to play another sport.
Bi-conditional- I am in athletics if and only if I play basketball. This is false, since you can be in basketball and not play basketball.
Since the conditional contains the hypothesis of being in athletics, this gives us multiple choices. You can be in numerous sports and be in athletics. Since the conclusion is playing basketball, the group of people in athletics is not selective to those in basketball. As a result all of these statements were false. In addition, every statement resulted in the possibility of their outcomes. If you weren't in athletics, you could still play basketball.
_________________________________________________________________________________________________________________________________________
2) There is a path crossing on a bike trail. Vinny, Aniket, Brice, and Dhruv decide to create a project regarding the angles that the trail intersect at. Vinny finds that QT and LP are perpendicular. Aniket, Dhruv, and Brice write down numbers, but they spill water on it so now they can't read it. However, they remember it was of the following equations. Aniket remembers that <LMN=2x+2y. Brice remembered that <NMQ=x+10. Lastly, Dhruv remembered that <TMR=3y. In order to find the angles they lost, they must find x and y. Help them find the values!
1) If I am in athletics, then I play basketball.
Using this conditional statement, show the 5 statements and their truth vlue. If it is false, give a counterexample.
Conditional- If I am in Athletics, then I play basketball. This is false as you can be in athletics and play either football or volleyball.
Converse- If I play basketball, then I am in athletics. This statement is also false, since you can play basketball for fun or play outside of school athletics.
Inverse- If I am not in Athletics, then I do not play basketball. This is a false conjecture, since I can still play basketball even though I am not in athletics.
Contrapositive- If I do not play basketball, then I am not in Athletics. This is false since I can be in Athletics even though I don't play basketball as I would be able to play another sport.
Bi-conditional- I am in athletics if and only if I play basketball. This is false, since you can be in basketball and not play basketball.
Since the conditional contains the hypothesis of being in athletics, this gives us multiple choices. You can be in numerous sports and be in athletics. Since the conclusion is playing basketball, the group of people in athletics is not selective to those in basketball. As a result all of these statements were false. In addition, every statement resulted in the possibility of their outcomes. If you weren't in athletics, you could still play basketball.
_________________________________________________________________________________________________________________________________________
2) There is a path crossing on a bike trail. Vinny, Aniket, Brice, and Dhruv decide to create a project regarding the angles that the trail intersect at. Vinny finds that QT and LP are perpendicular. Aniket, Dhruv, and Brice write down numbers, but they spill water on it so now they can't read it. However, they remember it was of the following equations. Aniket remembers that <LMN=2x+2y. Brice remembered that <NMQ=x+10. Lastly, Dhruv remembered that <TMR=3y. In order to find the angles they lost, they must find x and y. Help them find the values!
Solution: x=20, y=10
Since QT and LP are perpendicular, you know that <LMT and <LMQ are equal and both equal 90 degrees. Also you can't assume that NM bisects LMQ, which would make your answer wrong.
Since QT and LP are perpendicular, you know that <LMT and <LMQ are equal and both equal 90 degrees. Also you can't assume that NM bisects LMQ, which would make your answer wrong.
Since LMQ is 90 degrees, you can say that <LMN+<NMQ=<LMQ.
2x+2y+x+10=90 Given
3x+2y+10=90 Simplify
3x+2y=80 Subtraction POE
Also, since <NMQ and <TMR are a vertical angle pair, they are equal. <NMQ=<TMR.
x+10=3y Given
10=3y-x Subtraction POE
3y-x=10 Symmetric POE
Lastly, in order to find x and y you must use a system of equations.
2y+3x=80
+3(3y-x=10)
_____________
11y=110, Given
y=10 Division POE
Lastly, to find x, you must plug it back in.
If x+10=3y, then x+10=3(10)
x+10=3(10) Given
x+10=30 Distributive Property
x=20 Subtraction POE
In conclusion, x=20 and y=10. If you wanted you could go the next step and find the angles. <LMN=2x+2y, 2(20)+2(10)=60 degrees
<NMQ=x+10, <NMQ=20+10=30
<TMR=3y, <TMR=3(10)=30
2x+2y+x+10=90 Given
3x+2y+10=90 Simplify
3x+2y=80 Subtraction POE
Also, since <NMQ and <TMR are a vertical angle pair, they are equal. <NMQ=<TMR.
x+10=3y Given
10=3y-x Subtraction POE
3y-x=10 Symmetric POE
Lastly, in order to find x and y you must use a system of equations.
2y+3x=80
+3(3y-x=10)
_____________
11y=110, Given
y=10 Division POE
Lastly, to find x, you must plug it back in.
If x+10=3y, then x+10=3(10)
x+10=3(10) Given
x+10=30 Distributive Property
x=20 Subtraction POE
In conclusion, x=20 and y=10. If you wanted you could go the next step and find the angles. <LMN=2x+2y, 2(20)+2(10)=60 degrees
<NMQ=x+10, <NMQ=20+10=30
<TMR=3y, <TMR=3(10)=30
Dhruv's Practice Problems
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