Consider the diagram and the paragraph proof below. Given: Right △ABC as shown where CD is an altitude of the triangle Prove: a2 + b2 = c2 Because △ABC and △CBD both have a right angle, and the same angle B is in both triangles, the triangles must be similar by AA. Likewise, △ABC and △ACD both have a right angle, and the same angle A is in both triangles, so they also must be similar by AA. The proportions and are true because they are ratios of corresponding parts of similar triangles. The two proportions can be rewritten as a2 = cf and b2 = ce. Adding b2 to both sides of first equation, a2 = cf, results in the equation a2 + b2 = cf + b2. Because b2 and ce are equal, ce can be substituted into the right side of the equation for b2, resulting in the equation a2 + b2 = cf + ce. Applying the converse of the distributive property results in the equation a2 + b2 = c(f + e).
Answers
This is because since f + e = c,
then a² + b² = c(f + e)
a² + b² = c(c)
a² + b² = c²
This is the correct answer
Answer:
Because f+e=c
Therefore ,[tex]a^2+b^2=c^2[/tex]
Step-by-step explanation:
Given A right triangle ABC as shown in figure where CD is an altitude of the triangle.
To prove that [tex]a^2+b^2=c^2[/tex]
Proofe: Given [tex]\triangle ABC\; and\; \triangle CBD[/tex] both are right triangle and both triangles have common angle B si same.
Therefore , two angles of two triangles are equal .
Hence, [tex]\triangle ABC \sim\triangle CBD[/tex] by using AA similarity.
Similarity property: when two triangles are similar then their corresponding angles are equal and their corresponding side are in equal proportion.[tex]\frac{a}{f} =\frac{c}{a}[/tex]Similarly , [tex]\triangle ABC \sim \triangle ACD[/tex] by AA similarity property . Because both triangles are right triangles therefore, one angle of both triangles is equal to 90 degree and both triangles have one common angle A is same .
[tex]\therefore[/tex] [tex]\frac{b}{c} =\frac{e}{b}[/tex]
The corresponding parts of two similar triangles are in equal proportion therefore , two proportion can be rewrite as
[tex]a^2=cf[/tex] (I equation)
and [tex]b^2=ce[/tex] (II equation)
Adding [tex]b^2[/tex] to both sides of firs equation
[tex]a^2+b^2=cf+b^2[/tex]
Because [tex]b^2=ce[/tex] and ce can be substituted into the right side of equation wevcan write as
[tex]a^2+b^2=ce+cf[/tex]
Applying the converse of distributive property we can write
[tex]a^2+b^2=c(f+e)[/tex]
Distributive property: a.(b+c)= a.c+a.b
[tex]a^2+b^2=c^2[/tex]
Because f+e=[tex]c^2[/tex]
Hence proved.
Related Questions
how to discriminant of each quadratic equation then state the number and type of solutions. yall fr help me
Answers
Note, 1. the rules marked with blue colour, the solutions marked by red; 2. solutions are given shortly.
Final answer:
The discriminant of a quadratic equation determines the number and type of solutions.
Explanation:
The discriminant of a quadratic equation can help determine the number and type of solutions.
If the discriminant is greater than zero, there are two real solutions.
If the discriminant is zero, there is one real solution.
If the discriminant is negative, there are no real solutions.
A grandmother deposited $4,000 in an account that pays 5% per year compounded annually when her granddaughter was born. What will the value of the account be when the granddaughter reaches her 13th birthday?
Answers
A = 4000(1 + 0.05/1)^(13)(1)
A = $7542.59
The value of the account is $7542.59.
What is interest?It is computed by adding the annual interest rate raised to the number of compound periods and subtracting one after multiplying the initial principal amount by one. The resulting value is subsequently deducted from the loan's entire original amount.
Given:
P=4000
r = 5%
t = 13 years
Now, using the formula
[tex]A = P( 1 + r/n)^{nt}[/tex]
A = 4000[tex](1 + 0.05/1)^{(13)(1)[/tex]
A= 4000[tex](1.05)^{13[/tex]
A = 4000 x 1.8856
A = $7542.59
Hence, The value of the account is $7542.59.
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Jennifer made 5 liters of punch for her party. Her brother made another 750 mL. If they combine the two batches, how many 180mL servings would they have? Would there be any punch left over? If so how much? Please respond with the answer of how much of the two batches combined and how many 180mL there are
Answers
1 liter----------------> 1000 ml
Jennifer punch is 5 liters----------------> 5*1000=5000 ml
Her brother punch-------------------------> 750 ml
[total two batches]=(5000+750)----------> 5750 ml
the answer part A) is 5750 ml
Part B) how many 180 mL servings would they have?
[180 mL servings]=5750/180=31.94-----------> 31
they would have 31 (180 ml servings)
punch left over=[5750-31*180]=170 ml
there would be 170 ml of punch left over
The cone in the diagram has the same height and base area as the prism. What is the ratio of the volume of the cone to the volume of the prism?
Answers
For your figures: 1/3.
Answer:
B. 1/3
I had this question on Plato this is the answer.
Mr. jones has two children. the older child is a girl. what is the probability that both children are girls?
Answers
Hope this helps!
Final answer:
Given that the older child is already a girl, the probability that both children of Mr. Jones are girls is 50%, assuming an equal chance of having a boy or a girl.
Explanation:
The question is asking about the probability that both children of Mr. Jones are girls, knowing that the older child is already a girl. Since the older child is a girl, we only need to consider the probability that the second child is a girl. In a typical scenario where the probability of having a boy or a girl is equal (assuming a 50% chance for each), the probability that the second child is a girl, given that the family already has one girl, is simply 0.5 or 50%. Therefore, in this case, the probability that both children are girls is 50%.
the price of a tablet was reduced from $250 to $200. By what percentage was the price of the tablet reduced? Please help me solve this.
Answers
it doesn't interfere with 200.
Could someone help me with this?
Answers
*Take note that 12x and 6x are put together because they are like terms.
For Steps 3 and 4, you will see that the addition property of equality was used in STEP 3. To keep the equation equal, you will add the same number on both sides.
STEP 4 uses Division property of Equality. Like Step 3, to keep both sides of the equation equal, you must divide both sides with the same number. It keeps the statement true by doing so.
STEP 4 and 5 uses transitive property if you examine both as a whole.
Transitive property assumes that if a = b and b = c, then a = c
If 18/18 (a) = 1 (b), and x (c) = 18/18(a) then, x (c) = 1 (b).
identify the roots of -3x^3-21x^2+72x+540=0. State the multiplicity of each root.
Answers
y = -3*(x - 5)*(x+6)^2
(x - 1) has a multiplicity of 1 and (x + 6) has a multiplicity of 2 <<<=== answer
The graph can be found at wolframalpha
google it.
Efetue as divisões X ao cubo+10x+29x+20:x ao quadrado+5x+4
Answers
x + 5
---------------------------------
x^2 + 5x + 4 | x^3 + 10x^2 + 29x + 20
- x^3 + 5x^2 + 4x
----------------------------
5x^2 + 25x + 20
- 5x^2 + 25x + 20
-------------------------
0
(x^3 + 10x^2 + 29x + 20) / (x^2 + 5x + 4) = x + 5
There are 50 identical apples that need to be distributed among alice, bob, and cathy. determine the numberof ways that this can be done such that each one of them gets at least 2 apples. as examples, possible distributions ofthe identical apples among alice, bob and cathy include (a, b,
c.= (46, 2, 2); (10, 10, 30); (9, 2, 39), etc., where a,b and c denote the number of apples allocated to alice, bob and cathy respectively.
Answers
On the surface of the moon, the value of g is 1.67 m/s2. What is the horizontal distance traveled during a 2.00-meter high jump if the jumper runs as at 20 miles per hour?
Answers
h= 1/2 gt^2
2= 1/2 * (1.67) t^2
t^2= 1.67
t= 1.29
If the speed is 20 mph and the jump is 2 second, the distance traveled would be:
20 miles/ hour * 1.29 second * (1 hour/3600second)= 0.007179 miles
If you need to convert it to meter then: 0.007179 mile * 1609.34meter/ mile= 11.55 meter
Where a is determined by the conditions x2 + y2 ≤ 1 and x + y ≥ 1?
Answers
The area of a circle is 225 pie square inches. Find the area of the sector whose central angle is 45°.
Answers
The central angle for whole circle is 360°, it means that angle 45° has 45/360=1/8 part of the circle.
If the area of the circle is 225π, then the area of the sector 45° is 225π/8≈28.125π (in.²)
answer: 28.125π in².
The area of the sector of the circle is 28.125π inches²
What is a Circle?
A circle is a closed two-dimensional figure in which the set of all the points in the plane is equidistant from a given point called “centre”. Every line that passes through the circle forms the line of reflection symmetry. Also, the circle has rotational symmetry around the centre for every angle
The total internal angle is 360° about the centre
The area of the sector of the circle = ( θ / 360° )πr²
The perimeter of circle = 2πr
The area of the circle = πr²
where r is the radius of the circle
Given data ,
Let the area of the circle be = A
The value of A = 225 inches²
Now , the area of the sector whose central angle is 45° is calculated by
The area of the sector of the circle = ( θ / 360° )πr²
Substituting the value for θ = 45° , we get
Area of the sector of the circle = ( 45/360 ) x 225
= ( 1/8 ) x 225
= 28.125 inches²
Therefore , the area of the sector is 28.125 inches²
Hence , The area of the sector of the circle is 28.125π inches²
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A student is raising money for a homeless shelter. A local business agrees to donate an additional 20% of what the student raises, while a local politician also agrees to donate an additional 10% of what the student raises. If the student raises $600.00, how much will be donated?
Answers
Answer: the answer is $780.00
Step-by-step explanation:
$600.00 x .20 = $120.00
$600.00 x .10 = $60.00
Total:
$600.00 + $120.00 + $60.00 = $780.00
Answer:
780
Step-by-step explanation:
Example: y = -2x and y = x + 3
does the point (0,4) make either equation true? substitute it in and find out
Answers
x = 0
y = 4
Substitute x with 0 in the equations and evaluate if it would make 4 as the result of y.
First equation
y = -2x
y = -2 × 0
y = 0
The result is not 4, therefore point (0,4) doesn't make the first equation true.
Second equation
y = x + 3
y = 0 + 3
y = 3
The result is not 4, therefore (0,4) doesn't make the second equation true.
Katherine practices skiing for 1 1/2 hours every day except Saturday and Sunday. She practices for four hours on Saturday and three hours on Sunday. How many hours does she practice in a week?
Answers
1 1/2 hours x 5 days = 7 1/2 hours
4+3 +7 1/2 = 14 1/2 hours per week
A population of 1200 leopards decreases by 12% per year. How many leopards will there be in the population after 6 years? Round your answer to the nearest whole number
Answers
Which type of deposit is paid in advance to protect landlords against nonpayment?
a.security deposit
b.special deposit
c.surity deposit
20 POINTS!! pleasee help!
Answers
For the graph x = -3 find the slope of a line that is parallel to it.
-3
-1/3
0
undefined
Answers
the slope of the line that is parallel to x=-3 is classified as undefined.
let's check the answer
1) take any two points of the line x = -3
point A (-3,0) point B (-3,5)
2) find the slope of the line x=-3
m=(5-0)/(-3+3)=5/0----------> is undefined
the answer is the option undefined
Answer:
undefined
Step-by-step explanation:
Henry sells rings for $8 each. His expenses are $1.50 per ring, plus $91 for supplies. How many rings does he need to sell for his revenue to equal his expenses?
Answers
Answer:
14 rings.
Step-by-step explanation:
in order to calculate how many rings does he have to sell to break even we first have to calculate the profit that he has on the sell of every ring from which he can take money to pay per supplies, so that would be the selling price minues expenses:
8-1.5=6.5
Now we have to divide the 91 in supplies by the profit:
91/6.5=14
So he has to sell 14 rings in order to break even between expenses and supplies.
Henry needs to sell approximately 14 rings to cover his expenses, which include $1.50 per ring and $91 for supplies, with each ring priced at $8.
To find out how many rings Henry needs to sell for his revenue to equal his expenses, you can set up an equation.
Let's denote the number of rings he needs to sell as "x."
Revenue from selling x rings = Expenses
Revenue is calculated as the price per ring multiplied by the number of rings sold, which is $8x.
Expenses consist of the cost per ring and the supplies cost, which is $1.50x + $91.
So, the equation becomes:
$8x = $1.50x + $91
Now, let's solve for x:
Subtract $1.50x from both sides of the equation:
$8x - $1.50x = $1.50x + $91 - $1.50x
$6.50x = $91
Now, divide both sides by $6.50 to find the value of x:
x = $91 / $6.50
≈ 14
Therefore, Henry needs to sell approximately 14 rings for his revenue to equal his expenses.
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Ms Cohen gave a riddle: A string is 2 meters longer than a rod. Half of the rod is 1 meter shorter than half of the string. Is this true or false?
Answers
y: length of the string.
x: length of the rod.
We have the following equation:
y = x + 2
Half of the rod is 1 meter shorter than half of the string
x/2 = y/2-1
Let's rewrite equation 1 to see if it's true:
y = x + 2
(1/2) * y = (1/2) * (x + 2)
(1/2) * y = (1/2) * x + 1
(1/2) * x = (1/2) * y-1
Answer
this is true
(1/2) * x = (1/2) * y-1
Meis mothers is 4 times as old as mei. Mei’s mother is twice as old as mei’s mother.the sum of the three ages of 117. How old is mei, her mother, and her grandmother?
Answers
Final answer:
Mei is 9 years old, her mother is 36 years old, and her grandmother is 72 years old, determined by defining variables for their ages and solving the equation based on their age relationships and the given total sum of their ages.
Explanation:
To solve the problem regarding the ages of Mei, her mother, and her grandmother, we can use a system of equations based on the information given. Let's define Mei's age as m, her mother's age as 4m (since her mother is 4 times Mei's age), and her grandmother's age as 8m (since the grandmother is twice as old as Mei's mother).
According to the problem, the sum of their ages is 117 years:
m + 4m + 8m = 117
Simplifying this equation gives us:
13m = 117
Next, we divide both sides by 13 to find m:
m = 117 / 13
m = 9
Now that we have Mei's age, we can easily calculate her mother's and grandmother's ages:
Mei's mother: 4m = 4(9) = 36 years oldMei's grandmother: 8m = 8(9) = 72 years oldTherefore, Mei is 9 years old, her mother is 36 years old, and her grandmother is 72 years old.
1 What is the solution to the system of equations?
3x-4y=16
2x+3y=5
A (–4, 1)
B(4, –1)
C(–4, –1)
D (4, 1)
2 what is the solution to the system of equations?
7x-2y=-24
5x+5y=15
A (2, –5)
B (–2, 5)
C (–2, –5)
D (2, 5)
Answers
2. The answer is B (-2, 5)
Answer:
The answer for the first question would be B: (4, –1)
Step-by-step explanation:
Last year, 120 new houses were built in a neighborhood. So far this year, 56 of them have been sold. How many new houses have been sold this year?
Answers
right?
what is the slope of the line represented by the equation below? y=-2/3x+3
Answers
Answer:
-2/3
Step-by-step explanation:
The slope is the coefficient of x: -2/3.
___
This tells you how much y changes when x changes by 1 unit. The slope is the change in y divided by the change in x.
Describe how to solve problems for the area of a reduced or enlarged figure when the scale factor and dimensions of the original figure are given.
Answers
To solve problems for the area of a reduced or enlarged figure, determine the scale factor, find the corresponding dimensions, and calculate the area using the appropriate formula.
Explanation:To solve problems for the area of a reduced or enlarged figure when the scale factor and dimensions of the original figure are given, follow these steps:
1. Determine the scale factor by comparing the dimensions of the original figure to the dimensions of the reduced or enlarged figure.
2. Use the scale factor to find the corresponding dimensions of the reduced or enlarged figure.
3. Calculate the area of the reduced or enlarged figure using the appropriate formula for the shape (e.g., length * width for a rectangle).
For example, if the original figure is a rectangle with dimensions of 4 inches by 6 inches, and the scale factor is 2, the corresponding dimensions of the reduced or enlarged figure would be 8 inches by 12 inches. The area of the reduced or enlarged figure would then be calculated as 8 inches * 12 inches = 96 square inches.
Sample Response: To solve problems for the area of a reduced or enlarged figure, first square the scale factor. Then use the given dimensions to find the area of the original figure. Multiply the square of the scale factor by the area of the original to get the reduced or enlarged figure.
The formula for the area, A, of a circle is A = πr2 , where r is the circle’s radius, and the formula for the circumference, C, of a circle is C = πd , where d is the circle’s diameter. Which statement is correct?
Answers
Answer:
The formula for the circumference, C, of a circle is C = πd , where d is the circle’s diameter.
Step-by-step explanation:
For a circulus with radius r, we have that:
The area of the circulus is:
[tex]A = \pi r^{2}[/tex]
That is, the radius is squared, not multiplied by 2, so the first statement is incorrect.
The circumference of the circulus is:
The derivative of the area, so
[tex]C = 2\pi r[/tex]
However
The radius if half the diameter. So
[tex]C = 2\pi \frac{d}{2}[/tex]
[tex]C = \pi d[/tex]
So the correct statement is:
The formula for the circumference, C, of a circle is C = πd , where d is the circle’s diameter.
Both statements are correct. The formula for the area of a circle is A = πr², where r is the radius, and the formula for the circumference of a circle is C = πd, where d is the diameter.
The formula for the area of a circle is given by A = πr², where A represents the area and r represents the radius of the circle. In this formula, the radius is squared and then multiplied by the mathematical constant π (pi) to calculate the area enclosed by the circle.
On the other hand, the formula for the circumference of a circle is given by C = πd, where C represents the circumference and d represents the diameter of the circle. In this formula, the diameter is multiplied by the mathematical constant π to calculate the total distance around the circle.
These formulas are derived from the properties of circles. The radius is the distance from the center of the circle to any point on its edge, while the diameter is the distance across the circle passing through its center. The circumference represents the boundary or perimeter of the circle, while the area represents the space enclosed by the circle.
It's important to note that the relationship between the radius and diameter is given by the equation d = 2r, where d represents the diameter and r represents the radius. This relationship allows us to interchangeably use the radius or diameter in the formulas for the area and circumference of a circle.
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Complete Question:
1. The formula for the area, A, of a circle is A = [tex]\pi r^2[/tex] , where r is the circle’s radius.
2. The formula for the circumference, C, of a circle is C = πd , where d is the circle’s diameter.
Which of the above statements is correct?
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The table below shows the number of marbles of different colors in a bag: Color of Marbles Number of Marbles Pink 5 Blue 2 Green 3 Ursula draws a marble from the bag randomly without looking. She then draws another marble from the bag without replacing the first one. Which expression shows the probability of drawing green marbles in both the trials? 3 over 10 multiplied by 2 over 9 3 over 10 multiplied by 2 over 10 3 over 10 added to 2 over 9 3 over 10 added to 2 over 10
Answers
Answer:
3/10 * 2/9
Step-by-step explanation:
The length of time that an auditor spends reviewing an invoice is approximately normally distributed with a mean of 600 seconds and a standard deviation of 100 seconds. what is the probability that an auditor will spend more than 12 minutes on an invoice?
Answers
Explanation:
Since 1 minute = 60 seconds, we multiply 12 minutes by 60 so that 12 minutes = 720 seconds. Thus, we're looking for a probability that the auditor will spend more than 720 seconds.
Now, we get the z-score for 720 seconds by the following formula:
[tex]\text{z-score} = \frac{x - \mu}{\sigma} [/tex]
where
[tex]t = \text{time for the auditor to finish his work } = 720 \text{ seconds} \\ \mu = \text{average time for the auditor to finish his work } = 600 \text{ seconds} \\ \sigma = \text{standard deviation } = 100 \text{ seconds}[/tex]
So, the z-score of 720 seconds is given by:
[tex]\text{z-score} = \frac{x - \mu}{\sigma} \\ \\ \text{z-score} = \frac{720 - 600}{100} \\ \\ \boxed{\text{z-score} = 1.2}[/tex]
Let
t = time for the auditor to finish his work
z = z-score of time t
Since the time is normally distributed, the probability for t > 720 is the same as the probability for z > 1.2. In terms of equation:
[tex]P(t \ \textgreater \ 720) \\ = P(z \ \textgreater \ 1.2) \\ = 1 - P(z \leq 1.2) \\ = 1 - 0.885 \\ \boxed{P(t \ \textgreater \ 720) = 0.115}[/tex]
Hence, there is 11.5% chance that the auditor will spend more than 12 minutes in an invoice.
The probability that an auditor will spend more than 12 minutes on an invoice is approximately 88.49%.
Calculate the probability an auditor will spend more than 12 minutes on an invoice using the Z-score formula for a normal distribution.
The probability that an auditor will spend more than 12 minutes on an invoice can be calculated using the Z-score formula for a normal distribution.
Calculate the Z-score: Z = (12*60 - 600) / 100 = -1.2.Find the probability corresponding to this Z-score using a standard normal distribution table. P(Z > -1.2) = 0.8849.Therefore, the probability that an auditor will spend more than 12 minutes on an invoice is approximately 88.49%.What is the better estimate for the height of a desk,1 meter or 1 kilometer
Answers
The best estimate for the height of a desk is a meter. One kilometer is equal to 1,000 metres. A desk is obviously not 1,000 metres high because, well, our roofs wouldn't be that high!