Andrew is writing a coordinate proof to show that the triangle formed by connecting the midpoints of the sides of an isosceles triangle is itself an isosceles triangle. He starts by assigning coordinates as given.
Enter the answers in the boxes to complete the coordinate proof.
P is the midpoint of DE. Therefore, the coordinates of P are ( answer, b ).
Q is the midpoint of DF. Therefore, the coordinates of Q are (3a, answer).
R is the midpoint of EF. Therefore, the coordinates or R are ( answer, answer)
The length of PR is √ a^2+b^2. The length of QR is √ a^2 +b^2.
Comparing the expressions for the lengths of PR and QR shows that the lengths are equal. therefore, △PQR is isosceles
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The area of triangle DEF = 4 ( area of triangle QRP) which is determined that by comparing the expression for the lengths of PR and QR.
What are congruent triangles?Two triangles are said to be congruent if their corresponding sides and angles are equal.
According to the data provided,
Triangle ΔDEF is an isosceles triangle having DE, EF, and FD sides.
Since DE = EF and P,R, and Q are the midpoints of the sides DF, EF, and DE,
We know from isosceles triangle theorems that the area of a triangle formed by uniting the midpoints of two isosceles triangles is one-fourth the area of the isosceles triangle.
As evidence, consider the following:
Because point Q is the halfway of DE, its coordinates are (a, b). (Because the midpoint of a line segment always has coordinates that are half the total of the coordinates of the end points.)
Because point R is the midpoint of FE, its coordinates are (3a,b).
The length of the base, DF, in triangle DEF is 4a and the height is 2b. As a result, its area is 4ab. (Because the area of a triangle equals 1/2 its base × height.)
The length of the base, QR, of the triangle QRP is 2a, while the height is b.
As a result, its area is ab.
Therefore, the area of triangle DEF is four times that of triangle QRP.
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Related Questions
Hey, I am looking for a answer to this, I need a simple answer! It is for 25 points! Please and thank you, enjoy your day!
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This net represents an unfolded box. 4 in. 4x5=7 WXI- 2 sov 6 in. n 1 yo 10 in. sa 14 Sau What is the total surface area, in square inches, of this box?
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The total surface area will be given by:
[tex]A=2A1+2A2+2A3[/tex]Where:
[tex]\begin{gathered} A1=40in^2 \\ A2=24in^2 \\ A3=6\cdot4=60in^2 \end{gathered}[/tex]Therefore:
[tex]\begin{gathered} A=2\left(40\right)+2\left(24\right)+2\left(60\right) \\ A=80+48+120 \\ A=248in^2 \end{gathered}[/tex]Answer:
248 in²
Juanita works at a telemarketing company. She makes 12 sales calls per hour. Employees are encouraged to make more than 480 calls per week. Juanita has already made 180 calls this week. How many more hours, x, does Juanita need to work this week to reach the weekly goal of sales calls?
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Using the mathematical operations of subtraction and division, Dan needs more than 25 hours to make more than 300 sales calls for this week to achieve the weekly goal.
What are the mathematical operations?Subtraction and division operations are two of the basic mathematical operations.
The subtraction operation involves operating on the minuend using the subtrahend to arrive at the difference.
In other words, the difference from a subtraction operation is the minuend minus the subtrahend.
The division operation is then applied to the difference above to arrive at the quotient.
The number of sales calls per hour = 12
The expected sales calls per week > 480
The number of sales calls already made by Juanita = 180
The more hours required to make more than 300, x, is 25 or more hours (480 - 180)/12
Thus, the number of more hours, using mathematical operations, that Juanita needs to reach the weekly sales calls goal is 25 or more hours.
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at this rate how many hours must Danielle babysit to earn $96
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4 ---> 48
x ---> 96
[tex]\begin{gathered} 4\times96=x\times48 \\ 384=48x \\ \frac{384}{48}=\frac{48x}{48} \\ x=8 \end{gathered}[/tex]answer: 8 hours
You are given the sample mean and the population standard deviation. Use this information to construct the 90% and 95%confidence intervals for the population mean Interpret the results and compare the widths of the confidence intervals. Ifconvenient, use technology to construct the confidence intervals.A random sample of 40 home theater systems has a mean price of $126.00. Assume the population standard deviation is$18.60Construct a 90% confidence interval for the population mean.
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Solution
Given the following below
[tex]\begin{gathered} \bar{x}=\text{ \$126}.00 \\ n=40 \\ \sigma=\text{ \$18.60} \\ At\text{ 90\% the critical z is }\pm1.645 \end{gathered}[/tex]To find the 90% confidence interval, the formula is
[tex]\bar{x}\pm z\frac{\sigma}{\sqrt[]{n}}[/tex]Substituting values into the formula above
[tex]126\pm1.645\frac{18.6}{\sqrt[]{40}}[/tex]Solve the above
[tex]\begin{gathered} 126+_{}1.645\frac{18.6}{\sqrt[]{40}}=130.84\text{ (two decimal places)} \\ 126-1.645\frac{18.6}{\sqrt[]{40}}=121.16\text{ (two decimal place)} \end{gathered}[/tex]At 90% confidence interval, the population mean will be between 121.16 and 130.84 (121.16, 130.84)
Which is equivalent to multiplying a number by 10 to the power of 4
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Answer: 10,000
Step-by-step explanation:
Dilate the figure by the scale factor. Then enterthe new coordinates.A(1,3)B(4,2)K=3A’([?],[]).B'([10]C'([],[:]C(1,-3)Enter
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The new coordinates after the dilation are as follows;
[tex]\begin{gathered} A^{\prime}\text{ (3,9)} \\ B^{\prime}\text{ (12,6)} \\ C^{\prime}\text{ (3,-9)} \end{gathered}[/tex]Here, we want to perform a dilation
Given a pre-image with coordinates (x,y) and a scale factor of k, the coordinates of the image will be;
[tex](x,y)\text{ = (kx,ky)}[/tex]Applying this to the given coordinates, we have;
[tex]\begin{gathered} A^{\prime}\text{ = }(3\times1,3\times3)\text{ = (3,9)} \\ B^{\prime}=\text{ (4}\times3,\text{ 2}\times3)\text{ = (12,6)} \\ C^{\prime}\text{ = (1}\times3,\text{ -3}\times3)\text{ = (3,-9)} \end{gathered}[/tex]if 2 blams equal 15 droms and 5 droms equals 28 klegs, how many klegs are in a blam?
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2 blams ----------------- 15 droms
5 droms ----------------- 28 klegs
1 blam ------------------ ? klegs
5 droms ------------------- 28 klegs
15 droms -------------------- x
x = (15 x 28) / 5
x = 84 klegs
2 blams ------------------------- 84 klegs
1 blam --------------------------- ? klegs
x = (1 x 84)/2
x = 42 klegs
There are 42 klegs in 1 blam
what is 126945 rounded to the nearest hundred thousand
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To the nearest hundred thousand, we must first determine the number in the hundred thousand position. This is the digit 1. Now consider the number next to it. This is 2. Since two is not up to 5, we would round off all other nuber to zero.
Hence to the nearest hundred thousand, 126,945 is 100,000
Multiply.Simplify your answer as much as possible. 3v5 .6y.2y4v4
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Solution
[tex]3v^5(6y)2y^4v^4[/tex]For this case we can do the following:
[tex](3\cdot6)\cdot(v^{5+4})\cdot(y^{1+4})=18v^9y^5[/tex]And then the solution for this case would be:
18v^9 y^5
Given the following frequency table of values, is the mean, median, or mode likely to be the best measure of the center for the data set?ValueFrequency271280290300310320330340350360370380391402411425432442453Select the correct answer below:ModeMeanMedian
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The mean (average) of a data set is found by adding all numbers in the data set and then dividing by the number of values in the set. The median is the middle value when a data set is ordered from least to greatest. The mode is the number that occurs most often in a data set.
On the given dataset, if we ignore the outlier of the dataset, all those measures are going to be similar, but since we have a high frequency on the middle of our dataset and the presence of an outlier value, the most likely to describe the center of the dataset is the mode.
3. In television shows you may watch, scientists may patterns of data (growth of zombies, bacteria, etc). Below is a table that shows the number of zombies over a 4 day period. Days 0 1 2 3 4 # of zombies 2 8 32 128 512 *** a) What is the initial value? b) ** What is the growth/ decay factor?
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We are given a table that shows the number of zombies over a 4 day period.
a) What is the initial value?
The initial value means an initial population that is the number of zombies at day 0
From the table, we see that the initial population is 2
Initial value = 2
b) What is the growth/decay factor?
The growth/decay factor can be found by finding the ratio of the consecutive terms
r = 512/128 = 4
r = 128/32 = 4
r = 32/8 = 4
r = 8/2 = 4
As you can see, we have got the same ratio.
Since the ratio is greater than 1 then it is a growth factor.
If it was less than 1 then it would have been a decay factor.
Therefore, the growth factor is 4
Initial value = 2
Growth factor = 4
III Comprehensive Mid term Review Part Question 2 of 60 (1 point) | Question Attempt: 1 of Unlimited 2 6 3 4 Convert 35°F to degrees Celsius. If necessary, round your answer to the nearest tenth of a degree. Here are the formulas. 5 C (F-32) 9 FC + 32 5 25 OF
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Here, we want to make a unit conversion
Since we want to move from Fahrenheit to celsius, we are going to use the first formula
We simply are going to substitute the value of F = 35 into the equation
Thus, we have it that;
[tex]\begin{gathered} C\text{ = }\frac{5}{9}(35-32) \\ C\text{ = }\frac{5\times3}{9} \\ \\ C\text{ =}\frac{5}{3} \\ C\text{ = 1.7 deg celsius} \end{gathered}[/tex]Marcia bought a dozen roses for $16.38 (including tax). If tax was 5%, what was the cost of the roses before tax?
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Tax amount = 5% of $16.38
[tex]\begin{gathered} =\frac{5}{100}\times16.38=0.819\approx0.82 \\ \text{The actual cost of the roses before tax = 16.38-0.82= \$15.56} \end{gathered}[/tex]the sum of negative three and a number, divided by seven, is negative two
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Answer:
-3+x/7=-2 is equation, 7 is the answer
Step-by-step explanation:
-3+x/7=-2 add 3 to both sides.
x/7=1 multiply 7 to both sides
x=7 this is the answer
If f(x)= x² + 5x and g(x) = 3x + 2, finda. (f+g)(2) by evaluating f(2) and g(2).f(2)= g(2)=ƒ(2) + g(2) =b. (f+g)(x)=c. Evaluate your formula from part b at x = 2 to verify your answer to part a.(f+g)(2)=
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Given:
f(x)= x² + 5x and g(x) = 3x + 2
Required:
To calculate
f(2)= g(2)=
(f+g)(x)=
(f+g)(2)=
Explanation:
a)-
[tex]\begin{gathered} f(2)=(x^2+5x)(2) \\ \\ =((2)^2+5(2)) \\ \\ =(4+10)=14 \\ \\ g(2)=(3x+2) \\ \\ =(3\times2+2) \\ \\ =(6+2)=(8) \end{gathered}[/tex][tex]\begin{gathered} f(2)+g(2) \\ \\ =(2^2+5\times2)+(3\times2+2) \\ \\ =(4+5\times2)+(3\times2+2) \\ \\ =(4+10)+(6+2) \\ \\ =(14)+(8) \\ \\ =22 \end{gathered}[/tex]b)-
[tex]\begin{gathered} f(x)+g(x) \\ \\ f(x)+g(x)=(x^2+5x)+(3x+2) \\ \\ determine\text{ the sign} \\ \\ x^2+5x+3x+2 \\ \\ \\ x^2+8x+2 \end{gathered}[/tex]c)-
[tex]\begin{gathered} (f+g)(x) \\ \\ (x^2+5x+3x+2)(2) \\ \\ (x^2+8x+2)(2) \\ \\ ((2)^2+8(2)+2) \\ \\ (4+16+2) \\ \\ (22) \end{gathered}[/tex]Required answer:
[tex]\begin{gathered} a)-22\text{ } \\ \\ b)-x^2+8x+2 \\ \\ c)-22 \end{gathered}[/tex]write the word sentence as an equationa number w divided by 5 equals 6
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a number : w
A number w divided by 5: w/5
equals (=) 6
[tex]\frac{w}{5}=6[/tex]3) A map of a rectangular park has a length of 4 inches and a width of 6 inches. It uses a scale of 1 inch for every 30 miles. What is the actual area of the park? Show how you know. (From Unit 1, Lesson 12)
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Given:
Length of the park, l=4 inches
Width of the park, w=6 inches.
Here, 1 inch is equal to the 30 miles.
To find the actual area of the park:
Area of the park is, A=lw
That is,
[tex]\begin{gathered} A=4\times6 \\ =24\text{ inches} \end{gathered}[/tex]In miles,
[tex]\begin{gathered} A=24\times30 \\ =720\text{ miles} \end{gathered}[/tex]Hence, the area of the park in miles is 720 miles.
help me with this question please
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Answer: 56
To solve this problem, we will use the formula for Combination which goes by
[tex]_{}C(n,r)=^nC_r=_nC_r=\frac{n!}{r!(n-r)!}[/tex]From the problem, we know that:
n = 8
r = 3
Since there are a total of 8 different movie soundtracks and we only need 3.
Substituting it on the formula,
[tex]C(n,r)=\frac{n!}{r!(n-r)!}[/tex][tex]C(8,3)=\frac{8!}{3!(8-3)!}[/tex][tex]C(8,3)=56[/tex]Therefore, there are 56 possible selections of movie soundtracks at the music store.
Find the distance between: (6,12) and (7,-4)
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The formula for determining the distance betwwen two points is expressed as
[tex]\begin{gathered} \text{Distance = }\sqrt{(y2-y1)^2+(x2-x1)^2} \\ \text{From the information given,} \\ x1\text{ = 6, x2 = 7} \\ y1\text{ = 12, y2 = - 4} \\ \text{Distance = }\sqrt{(-4-12)^2+(7-6)^2} \\ \text{Distance = }\sqrt{-16^2+1^2}=\sqrt{257} \\ \text{Distance = 16.03} \end{gathered}[/tex]The distance is 16.03
consider a right triangle with acute angle of measure 0 and cos 0 = 4/5 use Pythagorean identity to write an equation that can be used to find Sim 0 is ( + ( 4/5 = 1
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HELP PLEASEEEEE!!!!!!
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Negatives act the opposite for positives so be careful for this.
2) Create the first four terms based on the given recursive formulas below. Also determine if the
sequence you made is an arithmetic or geometric sequence:
G(1) = 18, G(n) = 2 · G(n − 1). H(1) = 3,H(n) = 5 · H(n − 1)
-
-
J(1) = 3, J(n) = J(n − 1) + 5 M(1) = 3, M(n) = 2 · (n − 1)
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The first 4 terms of given recursive formulas will be as follows
G(n) = 18,36,72,144 (geometric sequence)H(n) = 3, 15, 75, 325 (geometric sequence)J(n) = 3, 8, 13, 18 (arithmetic sequence)M(n) = 3, 6, 12, 24 (geometric sequence)Recursive Expression:The recursive expression provides two pieces of information:
the first term in the sequence.
the pattern rule that takes each term from the previous term.
As we have G(1) = 18, G(n) = 2 · G(n − 1).
It's A geometric sequence with r as 2
∴ G(2) = 2 · G(2 − 1) = 2 · G(1) = 2(18) = 36.
∴ G(3) = 2 · G(3 − 1) = 2 · G(2) = 2(36) = 72.
∴ G(4) = 2 · G(4 − 1) = 2 · G(3) = 2(72) = 144.
As we have H(1) = 3, H(n) = 5 · H(n − 1).
It's A geometric sequence with r as 5
H(2) = 5 · H(2 − 1) = 5 . H(1) = 5(3) = 15.
H(3) = 5 · H(3 − 1) = 5 . H(2) = 5(15) = 75.
H(4) = 5 · H(4 − 1) = 5 . H(3) = 5(75) = 325.
As we have J(1) = 3, J(n) = J(n − 1) + 5
It's an arithmetic sequence with d as 5
J(2) = J(2 − 1) + 5 = J(1) + 5 = 3+5 = 8.
J(3) = J(3 − 1) + 5 = J(2) + 5 = 8+5 = 13.
J(4) = J(4 − 1) + 5 = J(3) + 5 = 13+5 = 18.
As we have M(1) = 3, M(n) = 2 · M(n − 1)
It's A geometric sequence with r as 2
M(2) = 2 · M(2 − 1) = 2 · M(1) = 2(3) = 6.
M(3) = 2 · M(3 - 1) = 2 · M(2) = 2(6) = 12.
M(4) = 2 · M(4 - 1) = 2 · M(3) = 2(12) = 24.
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69 billion and 580,000,000÷4,570,000=
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69000000126.9
Which of the following are solutions to the equation below?
Check all that apply.
9x – 64 = 0
A. 8
B.
_ c.
C D.
3
F.
ო/დ
I
დო
3
J E. -8
დო
3
8
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Answer:
C -8/3
F 8/3
Step-by-step explanation:
9x² - 64 = 0
==> 9x² = 64
==> x² = 64/9
x = ±√(64/9) = ±√64/√9 = ±8/3
i) Hence solve cosec 3y ÷ cot 3y + tan 3y = 0.5 for 0
[tex] \leqslant y \leqslant \pi \: radians \: giving \: your \: your \: am \\ answer \: in \: terms \: of \: \pi[/tex]
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The value of y is π/9 or 7π/9 for the range 0 ≤ y ≤ π.
cosec 3y /( cot 3y + tan 3y) = 0.5
Trigonometry formula is:
cosec Ф = 1/ sin Ф
cot Ф = cos Ф/ sinФ
tan Ф = sinФ/ cos Ф
[tex]\frac{\frac{1}{sin 3y} }{\frac{cos 3y}{sin 3y} +\frac{sin 3y}{cos 3y} }[/tex] = 0.5
[tex]\frac{\frac{1}{sin 3y} }{\frac{cos 3y^{2} + sin 3y^{2} }{sin 3y cos 3y} }[/tex] = 0.5
sin² 3y + cos ² 3y = 1
1/ sin 3y × sin 3y cos 3y = 0.5
cos 3y = 0.5 = 1/2
cos π/ 3 = 1/2
For 0≤ y ≤π
0 ≤3y ≤ 3π
So 3y = 2kπ + π/3 , k = 0, 1
y = π/ 9 or 7π/ 9
Therefore the value of y is π/9 and 7π/ 9 for the given range.
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Step 2: In the question above, find (using the method of your choice—formula, or graphing calculator function, or Excel, or app) the value of the test statistic t, to two places after the decimal point.
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From the question given,
The formula to find the test statistics mean is given below as,
[tex]t=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt[]{n}}}[/tex]Where,
[tex]\begin{gathered} \bar{x}\text{ is the population mean} \\ \mu\text{ is the sample mean} \\ \sigma\text{ is the standard deviation } \\ n\text{ is the }sample\text{ size} \end{gathered}[/tex]Given the values of the above below,
[tex]\begin{gathered} \bar{x}=2.27\text{ mins} \\ \mu=2.98\text{ mins} \\ \sigma=0.98\text{ mins} \\ n=20 \end{gathered}[/tex]Substituting the values into the formula of test statistics t above,
[tex]\begin{gathered} t=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt[]{n}}} \\ t=\frac{2.27-2.98}{\frac{0.98}{\sqrt[]{20}}}=\frac{-0.71}{\frac{0.98}{4.472}}=-\frac{0.71}{0.219}=-3.242 \\ t=-3.24\text{ (two decimal places)} \end{gathered}[/tex]Hence, the test statistics, t = -3.24 ( two decimal places)
you are 37 years old and have accumulated $150,000 in your savings account. you intend to add a fixed amount each month for twenty years. for the first five (5) years you add $100 at the end of each month. then $ 200 at the end of the month for the remaining time. given that the account pays an interest rate of 6% per year compounded monthly, how much money will you have at age 58 in your savings account?
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At age 58, the investor will have (Future Value) $571,816.64 in their savings account after accumulating $150,000 and then adding $100 monthly for five years and $200 monthly for fifteen years.
How is the future value at age 58 computed?The future value at age 58 can be computed in two tranches.
The first step is to compute the future value using the present value of the accumulated funds in addition to the periodic monthly investments.
The second step computes the future value using the present value of the funds after 5 years in addition to the periodic monthly savings.
Future Value at the end of the first 5 years:N (# of periods) = 60 months (5 x 12)
I/Y (Interest per year) = 6%
PV (Present Value) = $150,000
PMT (Periodic Payment) = $100
FV = $209,304.53
Sum of all periodic payments = $6,000 (60 x $100)
Total Interest = $53,304.53
Future Value at the end of the remaining years:N (# of periods) = 180 months (12 x 15 years)
I/Y (Interest per year) = 6%
PV (Present Value) = $209,304.53
PMT (Periodic Payment) = $200
Results:
Future Value (FV) = $571,816.64
Sum of all periodic payments = $36,000 ($200 x 180 months)
Total Interest = $326,512.11
Thus, you will have $571,816.64 as the future value in your savings account when you turn 58.
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A triangular priam has the following dimensions: width is 5 mm base is15 mm. What additional dimensions are needed to find the surface area ofthe prism?
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If we need to determine the surface area of a prims we would have to add the areas of each of the faces of the prism, therefore, we would also require to know the length of the prism.
Question
Marco is writing a coordinate proof involving a right isosceles triangle. Marco places his triangle on the coordinate plane such that the base of the triangle lies along the x-axis.
What coordinates should he assign to this third vertex of the isosceles triangle?
Responses
(2a, 2b)
begin ordered pair 2 a comma 2 b end ordered pair
(a, b)
begin ordered pair a comma b end ordered pair
(b, a)
begin ordered pair b comma a end ordered pair
(b, b)