Challenge A family wants to rent a car to go on vacation. Company A charges $75.50 and14¢ per mile. Company B charges $30.50 and 9¢ per mile. How much more does Company Acharge for x miles than Company B?For x miles, Company A charges dollars more than Company B.(Simplify your answer. Use integers or decimals for any numbers in the expression.)
Company A charges $75.50 and 14¢ per mile, then for x miles, Company A charges 75.5 + 0.14x dollars
Company B charges $30.50 and 9¢ per mile, then for x miles, Company B charges 30.5 + 0.09x dollars
Subtracting the second equation to the first one,
75.5 + 0.14x
-
30.5 + 0.09x
----------------------
45 + 0.05x
For x miles, Company A charges 45 + 0.05x dollars more than Company B.
Which of the following steps were applied to ABC obtain A’BC’?
Given,
The diagram of the triangle ABC and A'B'C' is shown in the question.
Required:
The translation of triangle from ABC to A'B'C'.
Here,
The coordinates of the point A is (2,5).
The coordinates of the point A' is (5,7)
The translation of the triangle is,
[tex](x,y)\rightarrow(x+3,y+2)[/tex]Hence, shifted 3 units right and 2 units up.
A decorator creates a scale drawing of a dinning room table. The length of the scale is 3 centimeters. The image represents the dimensions of the actual dinning room table. What is the area of the scale drawing?
From the image given, the dinning room table is a rectangle.
Given:
Length in inches = 90 inches
Width in inches = 45 inches
The scale of the length is 3 centimeters.
Now, let's find the scale of the table:
[tex]\frac{90}{3}=30\text{inches}[/tex]This means that 30 inches represents 1 centimeter.
Also, let's find the width in centimeters:
[tex]\frac{45}{30}=1.5\operatorname{cm}[/tex]Thus, we have:
Length of scale drawing = 3 cm
Width of scale = 1.5 cm
To find the Area of the scale drawing, use the area of a rectangle:
A = Length x Width
[tex]A=3\times1.5=4.5\operatorname{cm}^2[/tex]Therefore, the length of the scale drwing is = 4.5 cm²
ANSWER:
[tex]4.5\operatorname{cm}^2[/tex]Need help ASAP Which graph shows the asymptotes of the function f(x)= 4x-8 _____ 2x+3
First we will calculate the vertical asymptote, is when the denominator of the function given is equal to zero
[tex]\begin{gathered} 2x+3=0 \\ x=-\frac{3}{2} \end{gathered}[/tex]then we will calculate the horizontal asymptote because the degree of the numerator and the denominator is equal we can calculate the horizontal asymptote with the next operation
[tex]y=\frac{a}{b}[/tex]a= the coefficient of the leading term of the numerator
b=the coefficient of the leading term of the denomintor
in our case
a=4
b=2
[tex]y=\frac{4}{2}=2[/tex][tex]y=2[/tex]As we can see the graph that shown the asymptotes of the function is the graph in the option C.
What is an equation of a parabola with the given vertex and focus? vertex: (-2, 5)focus: (-2, 6)show each step
Explanation
the equation of a parabola in vertex form is give by:
[tex]\begin{gathered} y=a(x-h)^2+k \\ \text{where} \\ (h,k)\text{ is the vertex} \\ and\text{ the focus is( h,k}+\frac{1}{4a}) \end{gathered}[/tex]Step 1
so
let
a) vertex
[tex]\begin{gathered} vertex\colon(h.k)\text{ }\rightarrow(-2,5) \\ h=-2 \\ k=5 \end{gathered}[/tex]and
b) focus
[tex]\begin{gathered} \text{( h,k}+\frac{1}{4a})\rightarrow(-2,6) \\ so \\ h=-2 \\ \text{k}+\frac{1}{4a}=6 \\ \end{gathered}[/tex]replace the k value and solve for a,
[tex]\begin{gathered} \text{k}+\frac{1}{4a}=6 \\ 5+\frac{1}{4a}=6 \\ \text{subtract 5 in both sides} \\ 5+\frac{1}{4a}-5=6-5 \\ \frac{1}{4a}=1 \\ \text{cross multiply } \\ 1=1\cdot4a \\ 1=4a \\ \text{divide both sides by }4 \\ \frac{1}{4}=\frac{4a}{4}=a \\ a=\text{ }\frac{1}{4} \end{gathered}[/tex]Step 2
finally, replace in the formula
[tex]\begin{gathered} y=a(x-h)^2+k \\ y=\frac{1}{4}(x-(-2))^2+5 \\ y=\frac{1}{4}(x+2)^2+5 \\ \end{gathered}[/tex]therefore, the answer is
[tex]y=\frac{1}{4}(x+2)^2+5[/tex]I hope this helps you
For a period of d days an account balance can be modeled by f(d) = d^ 3 -2d^2 -8d +3 when was the balance $38
Given a modelled account balance for the period of d days as shown below:
[tex]\begin{gathered} f(d)=d^3-2d^2-8d+3 \\ \text{where,} \\ f(d)\text{ is the account balance} \\ d\text{ is the number of days} \end{gathered}[/tex]Given that the account balance is $38, we would calculate the number of days by substituting for f(d) = 38 in the modelled equation as shown below:
[tex]\begin{gathered} 38=d^3-2d^2-8d+3 \\ d^3-2d^2-8d+3-38=0 \\ d^3-2d^2-8d-35=0 \end{gathered}[/tex]Since all coefficients of the variable d from degree 3 to 1 are integers, we would apply apply the Rational Zeros Theorem.
The trailing coefficient (coefficient of the constant term) is −35.
Find its factors (with plus and minus): ±1,±5,±7,±35. These are the possible values for dthat would give the zeros of the equation
Lets input x= 5
[tex]\begin{gathered} 5^3-2(5)^2-8(5)-35=0 \\ 125-2(25)-40-35=0 \\ 125-50-75=0 \\ 125-125=0 \\ 0=0 \end{gathered}[/tex]Since, x= 5 is a zero, then x-5 is a factor.
[tex]\begin{gathered} d^3-2d^2-8d-35=(d-5)(d^2+3d+7)=0 \\ (d-5)(d^2+3d+7)=0 \\ d-5=0,d^2+3d+7=0 \\ d=0, \end{gathered}[/tex][tex]\begin{gathered} \text{simplifying } \\ d^2+3d+7\text{ would give} \\ d=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ a=1,b=3,c=7 \end{gathered}[/tex][tex]\begin{gathered} d=\frac{-3\pm\sqrt[]{3^2-4\times1\times7}}{2\times1} \\ d=\frac{-3\pm\sqrt[]{9-28}}{2} \\ d=\frac{-3\pm\sqrt[]{-17}}{2} \end{gathered}[/tex]It can be observed that the roots of the equation would give one real root and two complex roots
Therefore,
[tex]d=5,d=\frac{-3\pm\sqrt[]{-17}}{2}[/tex]Since number of days cannot a complex number, hence, the number of days that would give a balance of $38 is 5 days
The elimination method is used in place over substitution when one equation is not easily solved for ______________ variable.A) a standardB) a dependentC) an independentD) a single
Given:
There are given the statement about the elimination method and substitution method.
Explanation:
According to the concept:
One equation cannot be easily solved for a single variable.
Final answer:
Hence, the correct option is D.
Fill In the proportion No explanation just need answer got disconnected from last tutor
Explanation
Since the given shapes are similar, which implies that they are proportional,
Therefore; we will have
Answer:
[tex]\frac{AB}{EF}=\frac{BC}{FG}[/tex]A bookshelf holds 5 novels, 4 reference books, 3 magazines, and 2 instruction manuals.
Teacher example 1: In how many ways can you choose one reference book or one instructional manual?
# of reference books + # of instructional manual - # of options that are both 4 + 2 Ways to choose a reference book OR an instruction manual?
You try: In how many ways can you choose a magazine or a reference book? # of magazine + # of reference book - # of options that are both mag and reference book
Ways to choose a magazine or a reference book?
This is so confusing to me. any help would be amazing, 100 points!! help as soon as possible
We can choose one reference book or one instructional manual from the bookshelf in 48 different ways.
Given,
Number of novels = 5
Number of reference books = 4
Number of magazines = 3
Number of instruction manuals = 2
Total number of books = 5 + 4 + 3 + 2 = 14 books
We have to find the number of ways of choosing one reference book or one instructional manual.
Number of ways = 4! x 2!
Number of ways = 24 x 2
Number of ways = 48
That is,
We can choose one reference book or one instructional manual from the bookshelf in 48 different ways.
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Perform the indicated operation and write the answer in the form A+Bi
The Solution:
Given:
[tex](3+8i)(4-3i)[/tex]We are required to simplify the above expression in a+bi form.
Simplify by expanding:
[tex]\begin{gathered} (3+8i)(4-3i) \\ 3(4-3i)+8i(4-3i) \\ 12-9i+32i-24(-1) \end{gathered}[/tex]Collecting the like terms, we get:
[tex]\begin{gathered} 12-9i+32i+24 \\ 12+24-9i+32i \\ 36+23i \end{gathered}[/tex]Therefore, the correct answer is [option 3]
A bag of tokens contains 55 red, 44 green, and 55 blue tokens. What is the probability that a randomly selected token is not red? Enter your answer as a fraction.
Explanation
In the bag of tokens, we are told 55 red, 44 green, and 55 blue tokens. Therefore, the total number of tokens in the bag is
[tex]55+44+55=154[/tex]Hence to find the probability that a randomly selected token is not red becomes;
[tex]Pr(not\text{ red black})=\frac{n(green)+n(blue)}{n(tokens)}=\frac{44+55}{154}=\frac{99}{154}=\frac{9}{14}[/tex]Answer: 9/14
? Question
Rachel and Jeffery are both opening savings accounts. Rachel deposits $1,500 in a savings account that earns 1.5% interest,
compounded annually. Jeffery deposits $1,200 in a savings account that earns 1% interest per year, compounded
continuously.
If y represents the account balance after t years, which two equations form the system that best models this situation?
For the conditions stated, y=1500+2250t and y=1200+1200t, respectively, will be necessary equations because both Rachel and Jeffery are opening savings accounts. Rachel places $1,500 in a savings account that accrues annual compound interest of 1.5%. Jeffery places $1,200 in a savings account that accrues continuously compounded interest of 1% per year.
What is equation?A mathematical statement known as an equation is made up of two expressions joined together by the equal sign. A formula would be 3x - 5 = 16, for instance. When this equation is solved, we discover that the value of the variable x is 7.
Here,
according to given condition,
y=1500+1500*1.5t
y=1500+2250t
y=1200+1200*1t
y=1200+1200t
So the required equation will be y=1500+2250t and y=1200+1200t for the conditions given as Rachel and Jeffery are both opening savings accounts. Rachel deposits $1,500 in a savings account that earns 1.5% interest, compounded annually. Jeffery deposits $1,200 in a savings account that earns 1% interest per year, compounded continuously.
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please help me and answer quick because my brainly keeps crashing before i can see the answer
The surface area of a sphere is given by the formula
[tex]SA=4*pi*r^2[/tex]we have
r=24/2=12 ft ----> the radius is half the diameter
substitute
[tex]\begin{gathered} SA=4*pi*12^2 \\ SA=576pi\text{ ft}^2 \end{gathered}[/tex]word problems 1. Jackson spent $4.65 on popcorn and $2.83 on a soda while at the movies. How much more money did Jackson spend on popcorn than on soda? Jackson spent $ # # # more on popcorn than soda,
Find out the difference
so
(4.65-2.83)=$1.82
therefore
the answer is $1.82Referring to the figure, the polygons shown are similar. Findthe ratio (large to small) of their perimeters and areas.
SOLUTION
Consider the image below
The ratio of the side is given by
[tex]\begin{gathered} \text{large to small} \\ \frac{\text{large}}{small}=\frac{length\text{ of the side of the large triangle}}{Length\text{ of the side of small triangle }}=\frac{10}{5}=\frac{2}{1} \\ \\ \end{gathered}[/tex]Since the ratio of the side is the scale factor
[tex]\text{the scale factor =}\frac{2}{1}[/tex]hence The raio of the perimeters is the scale factor
Therefore
The ratio of their parimeter is 2 : 1
The ratio of the Areas is square of the scale factor
[tex]\text{Ratio of Area =(scale factor )}^2[/tex]
Hence
[tex]\begin{gathered} \text{ Since scale factor=}\frac{2}{1} \\ \text{Ratio of Area=}(\frac{2}{1})^2=\frac{2^2}{1^2}=\frac{4}{1} \\ \text{Hence} \\ \text{Ratio of their areas is 4 : 1} \end{gathered}[/tex]Therefore
The ratio of their Areas is 4 :1
How many ways can Rudy choose 4 pizza toppings from a menu of 16 toppings if each can only be chosen once
ANSWER:
1820 different ways
STEP-BY-STEP EXPLANATION:
We can use here combination rule for selection:
[tex]_nC_r=\frac{n!}{r!(n-r)!}[/tex]In this case n is equal to 16 and r is equal to 4, therefore, replacing and calculating the number in different ways, there:
[tex]\begin{gathered} _{16}C_4=\frac{16!}{4!(16-4)!}=\frac{16!}{4!\cdot12!} \\ \\ _{16}C_4=1820 \end{gathered}[/tex]So in total there are 1820 different ways Rudy can choose 4 pizza toppings.
translate the following into an equation:6 less decreased by twice a number results in 8
Let the number be x.
Twice the number means 2 * x = 2x
Twice the number decreased by 6 means
2x - 6
Given that the result is 8, we have
2x - 6 = 8
Which of the following ordered pairs is a solution to the graph of the system of inequalities? Select all that apply(5.2)(-3,-4)(0.-3)(0.1)(-4,1)
For this type of question, we should draw a graph and find the area of the common solutions
[tex]\begin{gathered} \because-2x-3\leq y \\ \therefore y\ge-2x-3 \end{gathered}[/tex][tex]\begin{gathered} \because y-1<\frac{1}{2}x \\ \therefore y-1+1<\frac{1}{2}x+1 \\ \therefore y<\frac{1}{2}x+1 \end{gathered}[/tex]Now we can draw the graphs of them
The red line represents the first inequality
The blue line represents the second inequality
The area of the two colors represents the area of the solutions,
Let us check the given points which one lies in this area
Point (5, -2) lies on the area of the solutions
∴ (5, -2) is a solution
Point (-3, -4) lies in the blue area only
∴ (-3, -4) not a solution
Point (0, -3) lies in the red line and the red line is solid, which means any point on it will be on the area of the solutions
∴ (0, -3) is a solution
Point (0, 1) lies in the blue line and the blue line is dashed, which means any point that lies on it not belong to the area of the solutions
∴ (0, 1) is not a solution
Point (-4, 1) lies on the area of the solutions
∴ (-4, 1) is a solution
The solutions are (5, -2), (0, -3), and (-4, 1)
find the value of x so that the function has the given value
j(x) = -4/3x + 7; j (x) = -5
Answer:
x = 13 [tex]\frac{2}{3}[/tex]
Step-by-step explanation:
j(x) = [tex]\frac{-4}{3}[/tex] x + 7 Substitute -5 for x
j(-5) = [tex]\frac{-4}{3 }[/tex] ( -5) + 7
or
j(-5) =[tex](\frac{-4}{3})[/tex] [tex](\frac{-5}{1})[/tex] + 7 A negative times a negative is a positive
j(-5) = [tex]\frac{20}{3}[/tex] + 7
j(-5) = [tex]\frac{20}{3}[/tex] + [tex]\frac{21}{3}[/tex] [tex]\frac{21}{3}[/tex] means the same thing as 7
j(-5) = [tex]\frac{41}{3}[/tex] = 13 [tex]\frac{2}{3}[/tex]
Where are all the tutors at??? Like it won’t even let me ask a tutor
The scatter plot is given and objective is to find the best line of fit for given scatter plot.
Let's take the few points of scatter plot,
(0,8) ,( -1,8) , (-4,10) ,( -8,12),(-10,14) (-12,14)
Take the line and check which graph contains most of the points of scatter plot.
[tex]1)\text{ f(x)=}\frac{-1}{2}x+8[/tex]The graph is ,
now take ,
[tex]2)\text{ f(x)=x+8}[/tex]The graph is,
This graph contains only one point of scatter plot.
Take,
[tex]3)\text{ f(x)= 10}[/tex]Now the take the last equation,
[tex]4)\text{ f(x)=-2x+14}[/tex]this graph contains no point of the scatter plot.
From all the four graph of the lines it is observed that option 1) is the best line of fit for given scatter plot. because it contains 3 points of scatter plotes . which is more than the other graph of line.
Answer: Option 1)
The mean amount of time it takes a kidney stone to pass is 16 days and the standard deviation is 5 days. Suppose that one individual is randomly chosen. Let X = time to pass the kidney stone. Round all answers to 4 decimal places where possible.a. What is the distribution of X? X ~ N(16Correct,5Correct) b. Find the probability that a randomly selected person with a kidney stone will take longer than 17 days to pass it. 0.2Incorrectc. Find the minimum number for the upper quarter of the time to pass a kidney stone. 0.8Incorrect days.
Answer:
• (a)X ~ N(16, 5)
,• (b)0.4207
,• (c)19.37 days
Explanation:
(a)
• The mean amount of time = 16 days
,• The standard deviation = 5 days.
Therefore, the distribution of X is:
[tex]X\sim N(16,5)[/tex](b)P(X>17)
To find the required probabability, recall the z-score formula:
[tex]z=\frac{X-\mu}{\sigma}[/tex]When X=17
[tex]z=\frac{17-16}{5}=\frac{1}{5}=0.2[/tex]Next, find the probability, P(x>0.2) from the z-score table:
[tex]P(x>0.2)=0.4207[/tex]The probability that a randomly selected person with a kidney stone will take longer than 17 days to pass it is 0.4207.
(c)The upper quarter is the value under which 75% of data points are found.
The z-score associated with the 75th percentile = 0.674.
We want to find the value of X when z=0.674.
[tex]\begin{gathered} z=\frac{X-\mu}{\sigma} \\ 0.674=\frac{X-16}{5} \\ \text{ Cross multiply} \\ X-16=5\times0.674 \\ X=16+(5\times0.674) \\ X=19.37 \end{gathered}[/tex]The minimum number for the upper quarter of the time to pass a kidney stone is 19.37 days.
If the cube root of D is equal to 4 , what is D equal to ?
Given:
The cube root of D = 4
so, we can write the following expression:
[tex]\sqrt[3]{D}=4[/tex]cube both sides to find d
So,
[tex]\begin{gathered} (\sqrt[3]{D})^3=4^3 \\ D=4\times4\times4 \\ \\ D=64 \end{gathered}[/tex]So, the answer will be D = 64
Find the surface area and the volume of the figure below round your answer to the nearest whole number
The shape in the questionis a sphere having
Radius = 10ft
Finding the Surface area
The surface area of a square is given as
[tex]\text{Surface Area of sphere = 4}\pi r^2[/tex]putting the value for radius
[tex]\begin{gathered} \text{Surface Area of sphere = 4 }\times\frac{22}{7}\times\text{ 10}\times10 \\ \text{Surface Area of sphere = }\frac{4\text{ }\times22\times10ft\times10ft}{7} \\ \text{Surface Area of sphere = }\frac{8800ft^2}{7} \\ \text{Surface Area of sphere = 1257.14ft}^2 \\ \text{Surface Area of sphere }\cong1257ft^2\text{ ( to the nearest whole number)} \end{gathered}[/tex]The surface area of the sphere = 1257 square feet
Finding the volume
The volume of a sphere is given as
[tex]\text{volume of sphere = }\frac{4}{3}\pi r^3[/tex]putting the value of radius
[tex]\begin{gathered} \text{Volume of sphere = }\frac{4}{3}\times\frac{22}{7}\text{ }\times10ft\text{ }\times10ft\text{ }\times10ft \\ \text{Volume of sphere = }\frac{88000ft^3}{21} \\ \text{Volume of sphere = 4190.47ft}^3 \\ \text{Volume of sphere}\cong4190ft^3\text{ (to the nearest whole number)} \end{gathered}[/tex]Therefore, the volume of the sphere = 4190 cubic feet
I need help with homework . BC=5, angle A=25 degree.
AC = 2.332
AB = 5.517
Explanation:
Given:
BC = 5.
Angle B = 25 degree.
Angle C = 90 degree.
The objective is to find AC and AB.
By the trigonometric functions, Consider AB as hypotenuse, AC as opposite and BC as adjacent.
Then, the relationship between opposite (AC) and adjacent (BC) cnbe calculated by trigonometric ratio of tan theta.
[tex]\begin{gathered} \tan \theta=\frac{opposite}{adjacent} \\ \tan 25^0=\frac{AC}{5} \\ AC=\tan 25^0\cdot5 \\ AC=2.332 \end{gathered}[/tex]Now, the length AB can be calculated by Pythagorean theorem,
[tex]\begin{gathered} AB^2=AC^2+BC^2 \\ AB^2=2.332^2+5^2 \\ AB^2=5.436+25 \\ AB^2=30.436 \\ AB=\sqrt[]{30.436} \\ AB=5.517 \end{gathered}[/tex]Let's check the value using trigonometric ratios.
For the relationship of opposite and hypotenuse use sin theta.
[tex]\begin{gathered} \sin \theta=\frac{opposite}{hypotenuse} \\ \sin 25^0=\frac{2.332}{y} \\ y=\frac{2.332}{\sin 25^0} \\ y=5.517 \end{gathered}[/tex]Thus both the answers are matched.
Hence, the length of the side AC = 2.332 and the length of the side AB = 5.517.
identify the amplitude and period of the function then graph the function and describe the graph of G as a transformation of the graph of its parent function
Given the function:
[tex]g(x)=cos4x[/tex]Let's find the amplitude and period of the function.
Apply the general cosine function:
[tex]f(x)=Acos(bx+c)+d[/tex]Where A is the amplitude.
Comparing both functions, we have:
A = 1
b = 4
Hence, we have:
Amplitude, A = 1
To find the period, we have:
[tex]\frac{2\pi}{b}=\frac{2\pi}{4}=\frac{\pi}{2}[/tex]Therefore, the period is = π/2
The graph of the function is shown below:
The parent function of the given function is:
[tex]f(x)=cosx[/tex]Let's describe the transformation..
Apply the transformation rules for function.
We have:
The transformation that occured from f(x) = cosx to g(x) = cos4x using the rules of transformation can be said to be a horizontal compression.
ANSWER:
Amplitude = 1
Period = π/2
Transformation = horizontal compression.
A recent survey asked respondents how many hours they spent per week on the internet. Of the 15 respondents making$2,000,000 or more annually, the responses were: 0,0,0,0,0, 2, 3, 3, 4, 5, 6, 7, 10, 40 and 70. Find a point estimate of thepopulation mean number of hours spent on the internet for those making $2,000,000 or more.
Given
The total frequency is 15 respondents
The responses were: 0,0,0,0,0, 2, 3, 3, 4, 5, 6, 7, 10, 40 and 70
Solution
The population mean is the sum of all the values divided by the total frequency .
[tex]undefined[/tex]sum 0f 5 times a and 6
Answer:
30a
Step-by-step explanation:
There is a bag filled with 5 blue and 4 red marbles.
A marble is taken at random from the bag, the colour is noted and then it is replaced.
Another marble is taken at random.
What is the probability of getting at least 1 blue?
The probability of getting exactly 1 blue marble from a bag which is filled with 5 blue and 4 red marbles is 40/81.
What is probability?Probability of an event is the ratio of number of favorable outcome to the total number of outcome of that event.
A bag is filled with 5 blue and 4 red marbles.
The total number of marble in the bag are,
5+4=9
One marble is taken at random from the bag, the color is noted and then it is replaced. The probability of getting blue marble is,
P(B)=5/9
probability of getting red marble is,
P(R)=4/9
The Probability of getting red marble in first pick and probability of getting blue marble in second pick
P1=5/9×4/9=20/81
The Probability of getting blue marble in first pick and probability of getting red marble in second pick is,
p2=4/9×5/9=20/81
The exactly 1 blue is taken out, when first marble is red and second is blue or the first one is blue and second one is red. Thus, the probability of getting exactly 1 blue is,
P=p1+p2
=20/81+20/81
40/81
Hence the probability of getting exactly 1 blue marble from a bag which is filled with 5 blue and 4 red marbles is 40/81.
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Use mental math to find all of the quotients equal to 50. Drag the correct division problems into the box.
4
,
500
÷
900
450
÷
90
45
,
000
÷
900
4
,
500
÷
90
450
÷
9
Quotients equal to 50
Answer: 45,000 ÷ 900=50
Step-by-step explanation:
In the given figure ABC is a triangle inscribed in a circle with center O. E is the midpoint of arc BC . The diameter ED is drawn . Prove that
Answer:
we can use two ways to write 180° along with the inscribed angle theorem to obtain the desired relation
Step-by-step explanation:
Given ∆ABC inscribed in a circle O where E is the midpoint of arc BC and ED is a diameter, you want to prove ∠DEA = 1/2(∠B -∠C).
SetupWe can add add arcs to make 180° in two different ways, then equate the sums.
arc EB +arc BA +arc AD = 180°
arc EC +arc CA -arc AD = 180°
Equating these expressions for 180°, we have ...
arc EB +arc BA +arc AD = arc EC +arc CA -arc AD
SolutionRecognizing that arc EB = arc EC, we can subtract (arc EB +arc BA -arc AD) from both sides to get ...
2·arc AD = arc CA -arc BA
The inscribed angle theorem tells us ...
arc AD = 2∠DEAarc CA = 2∠Barc BA = 2∠CMaking these substitutions into the above equation, we have ...
4∠DEA = 2∠B -2∠C
Dividing by 4 gives the relation we're trying to prove:
∠DEA = 1/2(∠B -∠C)