Let ABCD be a cyclic quadrilateral. Let P be the intersection of AD and BC, and let Q be the intersection of AB and CD. Prove that the angle bisectors of angle DPC and angle AQD are perpendicular.
Answers
Cyclic quadrilateral ABCD.
Produce AD and BC to meet at P.
Produce AB and CD to meet at Q,
Angle bisector PJ of angle DPC divides angle DPC into two equal parts,x.
Angle bisector QI of angle AQD divides angle AQD into two equal parts,y.
Solution:
Construct segment PQ.
Name U= angle BPQ
Name V=angle BQP.
See attached figure.
Steps:
1. Consider triangle BPQ:
U+V=angle ABP (exterior angle)........................(1)
2. Consider triangle DPQ
U+2X+V+2Y = 180-angle PDQ (exterior angle)
= angle ABC (cyclic quad->opposite angles are supplementary)
=>
U+2X+V+2Y = angle ABC...................................(2)
3. Add (1) and (2)
U+V + U+2X+V+2Y = angle DPQ+angle ABC
=>
2(U+V+X+Y)=180 degrees (angles ABP and ABC lie on a straight lie)
=>
U+V+X+Y = 90 degrees = angle IKJ
4. angle bisectors PJ and QI are perpendicular.
Related Questions
What is the probability that at least 2 students in a class of 36 have the same birthday?
Answers
[tex]p_{no} (n) = \frac{364}{365} \cdot \frac{363}{365} \cdot ... \cdot \frac{365-n+1}{365} = \frac{364!}{365^{n-1} (365-n)!} [/tex]
Therefore, the probabilty that there are at least 2 students with same birthday is the complementary of this value:
[tex]p_{yes} (n) = 1 -p_{no}(n) =1- \frac{364!}{365^{n-1} (365-n)!}[/tex]
And if we use n=36 (number of students), we find
[tex]p=1- \frac{364!}{365^{36-1} (365-36)!} \sim 0.83[/tex]
So, the probability that there are at least 2 students in a group of 36 with same birthday is approximately 83%.
Angela owns a set of 6-inch tall glasses. Each glass has a diameter of 3 inches. Angela has three people over for dinner. Each guest will be served a full glass of milk. How much milk will Angela need ?
Answers
Volume of 1 glass = (πd^2/4)*h = (π*3^2/4)*6 = 42.41 in^3
Each of the three guests will be served a glass full of milk.
Volume of milk required = 3*volume of 1 glass = 3*42.41 = 127.23 in^3
But,
1 in^3 = 0.0163871 liters
Therefore,
127.23 in^3 = 127.23*0.0163871 ≈ 2.09 liters will be required.
How does the graph of y=1/2(x+4)^2compare to the graph of y=(x-1)^2 -3?
Answers
[tex]\bf \bullet \textit{ stretches or shrinks horizontally by } A\cdot B\\\\ \bullet \textit{ flips it upside-down if } A\textit{ is negative}\\ ~~~~~~\textit{reflection over the x-axis} \\\\ \bullet \textit{ flips it sideways if } B\textit{ is negative}[/tex]
[tex]\bf ~~~~~~\textit{reflection over the y-axis} \\\\ \bullet \textit{ horizontal shift by }\frac{ C}{ B}\\ ~~~~~~if\ \frac{ C}{ B}\textit{ is negative, to the right}\\\\ ~~~~~~if\ \frac{ C}{ B}\textit{ is positive, to the left}\\\\ \bullet \textit{ vertical shift by } D\\ ~~~~~~if\ D\textit{ is negative, downwards}\\\\ ~~~~~~if\ D\textit{ is positive, upwards}\\\\ \bullet \textit{ period of }\frac{2\pi }{ B} [/tex]
and now with that template in mind,
bearing in mind the parent function for both is just y = x², or y = 1(1x + 0)² + 0.
[tex]\bf y=\stackrel{A}{\cfrac{1}{2}}(x\stackrel{C}{+4})^2[/tex]
is x² but expanded by twice as much vertically, and has a horizontal shift of +4, namely 4 units to the left.
y=(x-1)² - 3
is just x² but with a horizontal shfit of -1, namely 1 unit to the right, and a vertical shift of -3, namely 3 units down.
y = |x| translated two units to the right.
Answers
So we go from y = |x| to y = |x-2|
for her bedroom Marisol is buying a rug that is 6 feet by 8 feet her room is 9 feet by 11 feet how many square feet of floor will not be covered by Rug?
Answers
Final answer:
Marisol's bedroom is 99 square feet, and the rug area is 48 square feet. By subtracting the rug area from the total room area, we find that 51 square feet of floor will not be covered by the rug.
Explanation:
To find out how many square feet of Marisol's bedroom floor will not be covered by the rug, we must first calculate the area of her entire room and then the area of the rug. The area of a rectangle is found by multiplying the length by the width. The area of Marisol's room is 9 feet × 11 feet = 99 square feet. The area of the rug is 6 feet × 8 feet = 48 square feet.
Next, we subtract the area of the rug from the area of the entire room to find the uncovered floor area.
Uncovered floor area = Total room area – Rug area
= 99 square feet – 48 square feet
= 51 square feet
Therefore, 51 square feet of Marisol's bedroom floor will not be covered by the rug.
PLZ HURRRY!!
Of the 12,000 votes cast in an election, 42% of the votes were for Mr.Jones. How many votes were cast for all the other candidates?
Answers
= 5040
Therefore, Number of Voters left = 12000-5040
= 6960
2. An earthquake was felt throughout a circular area of 1,808.64 square miles. What was the radius of the circular area?
A. 24 miles B. 12 miles C. 576 miles D. 20 miles
Answers
Answer:
A. 24 miles
Step-by-step explanation:
Using 3.14 for π, we have ...
... Area = πr^2
... √(Area/π) = r = √(1808.64/3.14) = √576 = 24 . . . . miles
Answer:
its 24 miles so the correct answer would be A
Step-by-step explanation:
what is the solution -10 + sqrt x+8=-4
Answers
Add 10 to both sides to get:
√(x + 8) = 6
Square both sides to get our binomial out of the square root:
x + 8 = 6²
x + 8 = 36
Subtract 8 to both sides to get:
x = 28
Answer:
[tex]x=28[/tex]
Step-by-step explanation:
[tex]-10+\sqrt{x+8}=-4[/tex]
⭆ Add 10 to both sides:-
[tex]-10+\sqrt{x+8}+10=-4+10[/tex]
[tex]\sqrt{x+8}=6[/tex]
⭆ Square both sides:-
[tex]x+8=36[/tex]
⭆ Subtract 8 from both sides:-
[tex]x+8-8=36-8[/tex]
[tex]x=28[/tex]
OAmalOHopeO
Adams sister want to watch her intake of fat per day.he estimate that the dinner will have 48grams of fat.what percentage of her total daily allowance of fat will the dinner be round your answer to a whole number.the recommended daily allowance of fat is70g
Answers
Use the formula percent/of = x/100.
So, 48/70 and x/100 you cross multiply the numbers; 48*100 and 70*x and end up with 70x=4800.
Divide both sides by 70, which gives you 68.57.
Round the number up to 69 percent.
Can a chord also be classified as a radius? Explain your answer.
Answers
A chord is a straight sediment that joins two points of the circumference. The radius goes from the center of the circle, which is not on the radius.
PLZ HELP ASAP CIRCLES WILL GIVE BRANLIEST
Answers
If we are to look closely to the figure, angle I intercepted the arc BDR. Also, the angle D intercepted arc BIR. The sum of the measures these two arcs is 360°. Hence, the sum of the measures of the angles intercepting these arcs is 180°.
m∠I + m∠D = 180°
117° + m∠D = 180°
The measure of angle D is 63°.
Answer: 63°
what integer and-8 have the product of -96
Answers
12 times -8 = -96!
:) Have a nice day and maybe consider grading this answer brainliest? <3
12 * (- 8) = - 96.
A fluorite crystal is shaped like a regular octahedron with edges that are 2.1 inches. Which of these is the volume of the fluorite crystal in cubic inches? Round to the nearest hundredth
Answers
1. You have the following information given in the problem above: The regular octahedron with edges that are 2.1 inches.
2. Therefore, you must apply the formula for calculate the volume of a octahedron, which is:
V=(a^3√2)/3
3. Therefore, you have:
V=((2.1 inches)^3√2)/3
V=178.30 inches^3
To find the volume of the fluorite crystal, you can use the formula for the volume of an octahedron with the given edge length of 2.1 inches, rounded to the nearest hundredth to be approximately 7.69 cubic inches.
The volume of the fluorite crystal can be calculated as follows:
Identify the formula for the volume of an octahedron: V = (1/3) × [tex]\sqrt{2}[/tex] × a³, where 'a' is the edge length.
Substitute the given edge length: V = (1/3) × [tex]\sqrt{2}[/tex] × 2.1³.
Calculate the volume: V = 7.69 cubic inches (rounded to the nearest hundredth).
wanda rode her bike 21/10 miles. Which mixed number shows how far Wanda rode her bike?
Answers
Hope that helps
To convert the improper fraction 21/10 to a mixed number, divide 21 by 10 to get 2 with a remainder of 1. The mixed number is 2 1/10.
The student is asking for help converting an improper fraction to a mixed number. An improper fraction is one where the numerator is greater than the denominator. To convert 21/10 to a mixed number, we divide the numerator by the denominator.
Divide 21 by 10, which equals 2 with a remainder of 1.
The whole number part of the mixed number is the quotient, which is 2.
The remainder becomes the new numerator, and the original denominator stays the same, making it 1/10.
Therefore, the mixed number is 2 1/10.
Which point best approximates the root of 45
Answers
We rewrite 45 as 3²×5
thus
√45=√(3²×5)
pulling terms out from under the radical we obtain
3√5
thus the exact answer form is:
3√5
or the decimal form is:
6.70820393
A book club offers a choice of 8 books from a list of 40. In how many can a member make a collection?
Answers
This is a math combination problem. The number of ways a member can choose 8 books from a list of 40 is 76,904,685 ways.
Explanation:This is a combinatorics problem, specifically a combination problem. A combination is a selection of items from a larger set where the order does not matter. In this case, we have a set of 40 books and want to know how many ways one can select 8 books from those 40. The formula for calculating combinations is nCr = n! / [(n - r)! * r!] where 'n' represents the total number of items, 'r' represents the number of items to choose, 'nCr' represents the number of combinations of n items taken r at a time, and '!' denotes factorial, the product of all positive integers up to that number.
Applying the combination formula, a member can make the choice of 8 books from a list of 40 in:
40C8 = 40! / [(40 - 8)! * 8!] = 76,904,685 ways
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the number of bacteria in a container increases at the rate of R(t) bacteria per hour. If there are 1000 bacteria at time t=0 , which of the following expressions gives the number of bacteria in the container at time t=3 hours?
Answers
The expression for the total number of bacteria at t = 3 hours, given an initial 1000 bacteria at t = 0, is: [tex]\[ 1000 + \int_0^3 R(x) \, dx \][/tex]
To find the number of bacteria in the container at time t = 3 hours, given that the rate of increase is [tex]\( R(t) \)[/tex] bacteria per hour and there are 1000 bacteria at time t = 0, you can use the following steps:
Start with the initial condition :
You are given that there are 1000 bacteria at t = 0.
So, the initial condition is [tex]\( N(0) = 1000 \),[/tex] where N(t) represents the total number of bacteria at time t.
Integrate to find the change in number of bacteria over time :
Since the rate of increase is R(t), the total increase in bacteria from time t = 0 to t = t is given by integrating R(t) with respect to time.
This integral provides the cumulative increase in bacteria:
[tex]\[ \int_0^t R(x) \, dx. \][/tex]
Add the initial bacteria to the increase :
Since there are already 1000 bacteria at t = 0, the total number of bacteria at time t is given by:
[tex]\[ N(t) = 1000 + \int_0^t R(x) \, dx. \][/tex]
Find the expression for t = 3 hours :
To find the total number of bacteria at t = 3 hours, you substitute t = 3 into the expression derived in step 3:
[tex]\[ N(3) = 1000 + \int_0^3 R(x) \, dx. \][/tex]
Thus, the expression [tex]\( 1000 + \int_0^3 R(x) \, dx \)[/tex] gives the number of bacteria in the container at time t = 3 hours.
Question :
the number of bacteria in a container increases at the rate of R(t) bacteria per hour. If there are 1000 bacteria at time t=0 , which of the following expressions gives the number of bacteria in the container at time t=3 hours?
The correct expression that gives the number of bacteria in the container at time t = 3 hours is D. [tex]1000 + \int_0^3 R(t) \, dt[/tex]
The problem asks us to find the number of bacteria in a container at t = 3 hours, given that the number of bacteria increases at a rate of R(t) bacteria per hour and there are 1000 bacteria at time t = 0.
To solve this, we need to integrate the rate function R(t) to find the total change in the number of bacteria over the interval from t = 0 to t = 3.
1. Initial number of bacteria:
At t = 0, the number of bacteria is 1000.
2. Change in number of bacteria:
The rate of change of the number of bacteria is given by R(t) bacteria per hour.
To find the total number of bacteria added over the time interval from t = 0 to t = 3, we need to integrate R(t) from 0 to 3.
Total change in number of bacteria = [tex]\int_0^3 R(t) \, dt[/tex]
3. Total number of bacteria at t = 3 hours:
This is the sum of the initial number of bacteria and the total change in the number of bacteria over the 3 hours.
Number of bacteria at t = 3 hours = [tex]1000 + \int_0^3 R(t) \, dt[/tex]
Complete Question:
The number of bacteria in a container increases at the rate of R(t) bacteria per hour. If there are 1000 bacteria at time t=0, which of the following expressions gives the number of bacteria in the container at time t=3 hours?
(A) R(3)
(B) 1000+R(3)
(C) [tex]$\int_0^3 R(t) d t$[/tex]
(D) [tex]$1000+\int_0^3 R(t) d t$[/tex]
Question 19 find the solution
Answers
x= 5
Using the value of x in first equation we can write:
y = 6(5) + 1
y = 30 + 1
y = 31
So, the solution is (5, 31)
Thus, option 2nd is the correct answer
Polygon ABCDEF is the result of a translation of polygon GHIJKL . GL¯¯¯¯¯ is parallel to HI¯¯¯¯¯ in the pre-image. please help
Answers
GL parallel to HI in the pre-image ensures AB remains parallel to FE in the translation.
Given that GL and HI are parallel, we can infer that AB and FE will also remain parallel in the translation. The parallelism of lines GL and HI suggests that their corresponding sides, AB and FE, will exhibit the same parallel relationship during the translation process. This is a consequence of the consistent geometric transformation applied to all points on lines GL and HI. As each point on GL undergoes translation, its corresponding point on HI experiences the same displacement, preserving the parallelism.
Therefore, we can confidently state that AB and FE maintain their parallel orientation, aligning with the geometric properties of the translation that preserves parallel lines. The parallelism observed in the original configuration persists throughout the translation, ensuring that AB and FE remain parallel to each other.
For more such questions on parallel:
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Part A: Factor 2x2b2 + 5xb2 + 2b2. Show your work. (4 points) Part B: Factor x2 − 8x + 16. Show your work. (3 points) Part C: Factor x2 − 49. Show your work. (3 points)
Answers
[tex]x^2 - 8x + 16 = (x - 4) (x - 4) = (x-4)^2[/tex]
[tex]x^2 - 49 = x^2- 7^2 = (x - 7)(x + 7)[/tex]
Which five values would be used to graph a box-and-whisker plot for the following set of data? 36, 14, 27, 27, 33, 22, 41, 29
Answers
Q1 Q2 Q3
{14,22,27,(27}{,29),33,36,41}
add/div. by 2 add/div. by 2
28.5 34.5
Minimum: 14 (smallest number)
Quartile 1: 28.5 (middle number of q1)
Quartile 2: 28 (middle number) when you have and even amount of numbers you end up having 2 middle numbers, so you just add them then divide by 2. 27+29=56 56/2=28
Quartile 3: 34.5 (middle number of q3)
Maximum: 41 (biggest number)
I hope this helped you! :)
Answer:
Lower value: 14
Highest value: 41
Lower quartile: 24.5
Upper quartile: 34.5
Median: 28
Step-by-step explanation:
Explain why the ratio 3 feet / 1 yard has a value of one
Answers
Use the substitution method to determine whether the system of equations has one solution, no solution, or infinitely many solutions. 2x-y=10 x = y + 5
Answers
2x-y = 10, x = y + 5
To solve the system we substitute x = y + 5 in the first equation and it remains:
2 (y + 5) - y = 10
2y +10 -y = 10.
y = 0
Then we substitute y = 0 in any of the two equations to find x
Finally we have left that x = 5
Which of the following is a service that banks provide? A. Printing paychecks B. Insuring vehicles C. Access to cash D. Printing money
Answers
Answer:
C. Access to cash
Step-by-step explanation:
Bank provides access to cash in multiple manners, you can debit the money from your savings account, apply for different type of loans, draft your DD etc.
While for the other options like printing paychecks are done by any company in order to provided paychecks to its employees, Insurance companies provides vehicle insurance services to anyone will to get their vehicle insured and printing money is provided only by central bank of any country, not every bank is allowed to print money.
A pair of dice are rolled. What is the probability of rolling 4 or greater?
A: 5/36
B: 1/18
C: 1/36
D: 11/12
Answers
Explanation:
1) The sample space is formed by 36 pairs: 6×6 = 36.
2) Roolling 4 or more is the same that not rolling 3 or less
3) Rolling 3 or less are (1,1), (1,2), and (2,1). Those are 3 outcomes out of 36.
4) Therefore, not rolling 3 or less (the same that rolling 4 or more) has 36 - 3 = 33 outcomes out of 36.
5) The probability of 33 out of 36 is 33/36
6) Simplify the fraction 33/36 = 11/12
And that is the answer.
An item is called "price elastic" if you drop the price and revenue goes up. Last week you sold 10 sodas for $2 each; this week you sold 30 sodas for $1 each. Is soda price elastic?
Answers
Final answer:
Soda is price elastic because lowering the price from $2 to $1 resulted in an increase in total revenue from $20 to $30, indicating that a decrease in price caused a larger increase in quantity sold.
Explanation:
When determining whether an item is price elastic or not, we look at the effect that changing the price has on the total revenue. If a decrease in price leads to an increase in total revenue, the demand for the product is considered price elastic. In the example given, soda was sold at different prices and quantities: 10 sodas at $2 each last week, and 30 sodas at $1 this week. We use these numbers to calculate the total revenue for each week:
Last week: 10 sodas × $2/soda = $20
This week: 30 sodas × $1/soda = $30
Since the total revenue increased from $20 to $30 when the price was lowered from $2 to $1, we can see that the revenue increased as a result of the price drop, indicating that the demand for soda is indeed price elastic.
You put $400 in an account. The account earns $18 simple interest in 9 months. What is the annual interest rate?
Answers
thats the answer hope this helps
Answer:
6%
Step-by-step explanation:
Let's start by finding the amount earned per month:
[tex]\frac{18}{9}[/tex]=2
Now let's find the % per month
[tex]\frac{2}{400} =.005[/tex], or .5%
Now let's multiply by 12 months to get the annual interest rate
[tex].005 * 12 = .06[/tex] or 6%
simplify this problem
Answers
Rewrite the fraction as division:
[tex]= (49 - \dfrac{1}{r^2}) \div (7 - \dfrac{1}{r} )[/tex]
Make them into single fraction:
[tex]= \dfrac{49r^2 - 1}{r^2} \div \dfrac{7r - 1}{r} [/tex]
Change the divide fraction into multiplication fraction:
[tex]= \dfrac{49r^2 - 1}{r^2} \times \dfrac{r}{7r - 1} [/tex]
Factorise the difference of square a² - b² = (a + b) (a - b) :
[tex]= \dfrac{(7r+ 1)(7r - 1)}{r^2} \times \dfrac{r}{7r - 1} [/tex]
Cancel the common factors:
[tex]= \dfrac{(7r+ 1)}{r}[/tex]
Solution
[tex] \frac{49 - a^2}{7 - a} \\ \frac{(7 - a)(7 + a)}{7 - a}\text{ Notice a cancellation can take place} [/tex]
There is a 7 - a in both numerator and denominator, so that cancel providing a does not equal 7. a cannot equal 7 because that will put a 7 in the denominator and that makes the whole fraction = something over 0 which is undefined.
Answer
So far what we have is
7 + a
But a = 1/r
So the answer can be 7 + 1/r
r can be anything but 0 [for this answer]
and 1/7 for the cancellation.
What are the lateral area and the surface of the cone shown below?
Answers
[tex]\bf \textit{total surface area of a cone}\\\\ SA=\pi r\stackrel{slant~height}{\sqrt{r^2+h^2}}+\pi r^2~~ \begin{cases} r=radius\\ h=height\\ -------\\ r=11\\ \stackrel{slant~height}{\sqrt{r^2+h^2}}=24 \end{cases} \\\\\\ SA=\pi (11)(24)+\pi (11)^2\implies SA=264\pi +121\pi \\\\\\ \boxed{SA=385\pi }[/tex]
what is the surface area of a sphere with a radius of 3m