The radius of the circle drawn in a right-angled trapezoid is 4 cm, and the acute angle is equal to 30°. Calculate the length of the middle line.
Answers
In a right-angled trapezoid, the middle line is the average of the lengths of the two parallel sides. Let's call the two parallel sides of the trapezoid "a" and "b", and the length of the middle line "m".
Since the radius of the circle inscribed in the trapezoid is 4 cm, we know that the distance from the center of the circle to each side of the trapezoid is 4 cm. Let's call this distance "r".
We can draw a perpendicular from the center of the circle to one of the parallel sides of the trapezoid, as shown in the diagram below:
_______
/ \
/ \
/ r \
/____________\
a
Since the acute angle of the trapezoid is 30°, we know that the angle between the perpendicular and the shorter parallel side is 60° (since the two angles add up to 90°). Therefore, we can use trigonometry to find the length of the shorter parallel side:
tan(60°) = r/a
a = r/tan(60°)
a = 4/tan(60°)
a ≈ 4*1.732 ≈ 6.928
Similarly, we can draw a perpendicular from the center of the circle to the longer parallel side of the trapezoid, as shown in the diagram below:
_______
/ \
/ \
/ r \
/____________\
b
Since the two perpendiculars and the longer parallel side form a right triangle, we can use the Pythagorean theorem to find the length of the
b² = (2r)² - m²
But we don't know the length of the middle line yet. However, we can use the fact that the middle line is the average of the lengths of the two parallel sides:
m = (a + b)/2
b = 2m - a
Substituting this expression for b into the equation above, we get:
(2m - a)² = (2r)² - m²
4m² - 4am + a² = 4r² - m²
5m² - 4am + a² = 4r²
5m² - 4(4/tan(60°))m + (4²) = 4(4²)
5m² - 16(√3)m + 16 = 64
5m² - 16(√3)m - 48 = 0
We can solve this quadratic equation using the quadratic formula:
m = (-b ± sqrt(b² - 4ac))/2a
In this case, a = 5, b = -16(√3), and c = -48. Substituting these values into the quadratic formula, we get:
m = (-(-16(√3)) ± sqrt((-16(√3))² - 4(5)(-48)))/(2(5))
m = (16(√3) ± sqrt(256(3) + 960))/10
m = (16(√3) ± sqrt(1728))/10
m = (16(√3) ± 24)/10
We take the positive root since the length of the middle line must be positive. Therefore,
m = (16(√3) + 24)/10
m = (8(√3) + 12)/5
m ≈ 5.797
Now we can use this value of m to find the length of the longer parallel side:
b = 2m - a
b = 2(5.797) - 6.928
b ≈ 4.666
Therefore, the shorter parallel side has length a ≈ 6.928 cm and the longer parallel side has length b ≈ 4.666 cm.
Related Questions
50 POINTS!! BRAINLIEST TO THE FIRST ANSWER!!
Answers
Answer:
i really dont know but yeh
Answer:
7
Step-by-step explanation:
You bought a magazine for $5 and four erasers. You spent a total of $25. What equation
would you use to solve for the price of each eraser? Use x as the variable.
Answers
Answer:
$25 - $5 = $20
$20 divided by 4 = 5
1 earser = 5
Suppose -1 1 PLEASE PLEASE PLEASE PLEASE
Answers
What's the question?
I don't quite understand what you're asking.
solve the system of equations by the substitution method y= 2x + 15y - 6x = 13
Answers
We are asked to solve a system of equations using the substitution method:
y = 2 x + 1
5 y - 6 x = 13
So we use the first equation as our substitution for "y", since we know that y is equal to "2 x + 1"
We use this expression in the second equation replacing "y":
5 (2x + 1) - 6 x = 13
and now we solve for the only unknown "x" that was left in the equation. We use distributive property to get rid of the parenthesis:
10 x + 5 - 6 x = 13
we combine the terms in x and subtract 5 from both sides:
4 x = 13 - 5
4 x = 8
x = 8/4
x = 2
Now we use this result in the first equation (our substitution equation):
y = 2 (2) + 1 = 5
therefore, x = 2 and y = 5 are the solutions to the system, also written in coordiate pair form as: (2, 5).
What is the equation of the line that passes through the point (1,-5) and has a slope of -3?
Answers
The equatiοn οf the line that passes thrοugh the pοint (1,-5) and has a slοpe οf -3 is y = -3x - 2.
What is equatiοn?A fοrmula knοwn as an equatiοn uses the equals sign (=) tο express hοw twο expressiοns are equal.
The equatiοn οf a line in slοpe-intercept fοrm is y = mx + b, where m is the slοpe and b is the y-intercept. We are given that the line passes thrοugh the pοint (1,-5) and has a slοpe οf -3. Sο we can substitute these values intο the slοpe-intercept fοrm and sοlve fοr b:
-5 = (-3)(1) + b
-5 = -3 + b
b = -2
Nοw that we knοw the y-intercept, we can write the equatiοn οf the line in slοpe-intercept fοrm:
y = -3x - 2
Therefοre, the equatiοn οf the line that passes thrοugh the pοint (1,-5) and has a slοpe οf -3 is y = -3x - 2.
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Find the volume of the prism.
7 m
20 m
26 m
Answers
A parabola can be drawn given a focus of (8,10) and a directrix of y=6. write the equation of the parabola in any form
pls help its due in 5 min
Answers
The equation of the parabola is h²-16h+8k+32=0 when parabola can be drawn with a focus of (8,10) and a directrix of y=6.
Given that,
A parabola can be drawn with a focus of (8,10) and a directrix of y=6.
We have to find the equation of the parabola in any form.
We know that,
The distance of any point P is (h,k) on a parabola form a focus is equal to its perpendicular distance from the directrix.
So,
√((h-8)²+(k-2)²) = | (k-6)/√0²+1² |
Squaring on both sides
(h-8)²+(k-2)² = (k-6)²
From th formula (a-b)² = a²-2ab+b²
We get,
h²-16h+64+k²-4k+4 = k²-12k+36
h²-16h+64+k²-4k+4 - k²+12k-36 =0
h²-16h+64-4k+4+12k-36=0
h²-16h+8k+32=0
Therefore, The equation of the parabola is h²-16h+8k+32=0 when parabola can be drawn with a focus of (8,10) and a directrix of y=6.
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help!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Answers
Answer:
1st
Step-by-step explanation:
llean works as a welder in web designer as a welder she earns $18 per hour last week she works x hours as a wlder and her Fran paid her $75 to design a webpage.write an expression to represent the total amount llean earned last week
Answers
I wasn’t able to find the multiplication sign on my keyboard and didn’t want to confuse you so I replaced x with times
en el coliseo de una ciudad, se jugo la final de un campeonato de voley . En total , 1200 personas asistieron al coliseo . esta cantidad de personas representa a los 3/4 de su capacidad. ¿Cual es la capacidad que tiene este coliseo?
a)900
b)1200
c)1600
d)4800
Answers
c. 1600
1200 : 3 * 4
= 400 * 4
= 1600
The capacity of this coliseum is 1600
The correct option is (C)
What is fraction?A fraction is a part of a whole. In arithmetic, the number is expressed as a quotient, in which the numerator is divided by the denominator. In a simple fraction, both are integers. A complex fraction has a fraction in the numerator or denominator.
Given:
Total people attended the Coliseum= 1200
let he capacity be x
3/4*x= 1200
x= 1200*4/3
x= 1600
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Your question is incomplete/other language, probably the traslated question/missing part is:
In a city coliseum, the final of a volleyball championship was played. In total, 1,200 people attended the Coliseum. this number of people represents 3/4 of its capacity. What is the capacity of this coliseum?
If varies inversely as (x 2 )and y=16, then x = 5 , so find x & y = 100(hint y = k/ x 2 )
Answers
When y = 100, x is approximately equal to 0.04.
If y varies inversely as x^2 and y = 16 when x = 5, we can find the values of x and y when y = 100.
To solve this problem, we can use the inverse variation formula, which states that y = k/x^2, where k is the constant of variation.
Given that y = 16 when x = 5, we can substitute these values into the formula to find the value of k.
16 = k/(5^2)
16 = k/25
To find k, we can cross multiply:
16 * 25 = k
400 = k
Now that we know the value of k, we can use it to find the value of y when x = 100.
y = k/(100^2)
y = 400/(100^2)
y = 400/10000
y = 0.04
Therefore, when y = 100, x is approximately equal to 0.04.
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Mia bought 8 copies of the same book. She spent $32.
What was the cost of one book?
Answers
Answer:
rhe cost of one book was $4. 32/8=4
Step-by-step explanation:
32÷8=4 so mia spent 4 dollars
solve 2x-3=-4 solution to the system
Answers
Answer:
x=-1/2
Step-by-step explanation:
10.50 a plus 3.75 b equals 2071.50, where a is the number of adult tickets sold and b is the number of student tickets sold, can be used to find the number of adult and student tickets. If 82 students attended, how may adult tickets were sold?
Answers
The dimension of the row space of a 3 x 3 matrix A is 2. (a) What is the dimension of the column space of A? (b) What is the rank of A? (c) What is the nullity of A? (d) What is the dimension of the solution space of the homogeneous system Ax = 0?
Answers
a) the dimension of its column space is also 2. b) the rank of A is 2. c) the nullity of matrix A is 1. d) the dimension of the solution space of the homogeneous system \(A_x = 0\) is also 1.
(a) The dimension of the row space of a matrix is equal to the dimension of its column space. So, if the dimension of the row space of matrix A is 2, then the dimension of its column space is also 2.
(b) The rank of a matrix is defined as the maximum number of linearly independent rows or columns in the matrix. Since the dimension of the row space of matrix A is 2, the rank of A is also 2.
(c) The nullity of a matrix is defined as the dimension of the null space, which is the set of all solutions to the homogeneous equation Ax = 0. In this case, the matrix A is a 3 x 3 matrix, so the nullity can be calculated using the formula:
nullity = number of columns - rank
nullity = 3 - 2 = 1
Therefore, the nullity of matrix A is 1.
(d) The dimension of the solution space of the homogeneous system Ax = 0 is equal to the nullity of the matrix A. In this case, we have already determined that the nullity of matrix A is 1. Therefore, the dimension of the solution space of the homogeneous system \(A_x = 0\) is also 1.
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Can you help me............
Answers
Answer:
1,652,43
Step-by-step explanation:
The numbers of runs scored by a baseball team in a sample of five games were:Fill in the blanks
Answers
a) The mean can be calculated by adding all values and then divide by the total number of values:
\(\text{Mean}=\frac{3+1+0+0+6}{5}=2\)b) The median is the middle value in the list of numbers, then you have to order the numbers as follows: 0, 0, 1, 3, 6.
\(\text{Median}=\text{ 1}\)c) The mode is the value that occurs most often, as you have a 0 value twice, then
\(\text{Mode}=\text{ 0}\)d) To calculate the sample variance you can use this formula
\(\begin{gathered} s^2=\frac{\sum^{}_{}(x-\bar{x})^2}{n-1}\text{ where x is each value, }\bar{x}\text{ is the mean, and n the number of values} \\ s^2=\frac{(0-2)^2+(0-2)^2+(1-2)^2+(3-2)^2+(6-2)^2}{5-1} \\ s^2=\frac{(-2)^2+(-2)^2+(-1)^2+(1)^2+(4)^2}{4} \\ s^2=\frac{4+4+1+1+16}{4} \\ s^2=\frac{26}{4}=6.5 \end{gathered}\)The sample standard deviation is the square root of sample variance, then
\(\begin{gathered} s=\sqrt[]{s^2}=\sqrt[]{6.5} \\ s=2.55 \end{gathered}\)A number divided by -9 is -16
Answers
Answer:
144
Step-by-step explanation:
144/-9=-16 Hope this helps
EASY INEQUALITIES!!!!
Answers
Answer:
2.16, 3.45
Step-by-step explanation:
Answer:
Hello!!! erz here ^^
Step-by-step explanation:
2.16< \(\sqrt{5}\) < 3.45
Hope this helps!! :D
amelia sells jewelry. she earns 12% commission on her sales , and a weekly salary of 550. how much did amelia earn this month if she sold 22,540 worth of jewelry?
Answers
Answer:
$27,444.80
Step-by-step explanation:
12% of 22,540 is 2,704.8
add the amount she made
add the weekly salary (550 × 4)
(c) Using your answers from (a) and (b), determine the area of the shaded region.
Answers
The calculated value of the area of the shaded region is 3.4 square inches
Determining the area of the triangle ABCGiven that
Side lengths = 8
Vertex angle = 50
So, we have
Area = 1/2 * 8² * sin(50)
Evaluate
Area = 24.5
Determining the area of the circular sectorGiven that
Radius, r = 8
Vertex angle = 50
So, we have
Area = Angle/360 * πr²
Evaluate
Area = 50/360 * 22/7 * 8²
Evaluate
Area = 27.9
Determining the area of the shaded region.This is calculated as
Shaded region = 27.9 - 24.5
So, we have
Shaded region = 3.4
Hence, the area of the shaded region is 3.4 square inches
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Complete question
An isosceles triangle ABC is shown below with legs that measure 8 inches and a vertex angle of 50'.
(a) Determine the area of AABC.
(b) Determine the area of the circular sector.
(c) Using your answers from (a) and (b), determine the area of the shaded region.
−6⋅f(3)−6⋅g(−1)=
Try to give an answer that's suitable for Khan Academy, thank you.
Answers
Answer:
-24
Step-by-step explanation:
Answer:
Step-by-step explanation:
-24
Points R, L, and S are
A. non-collinear and non-coplanar
B. coplanar and collinear
C. non-collinear and coplanar
D. collinear and collinear
Answers
Answer:
C. non collinear and coplanar
Option (C) non-collinear and co planar is the correct answer about the points R, L, and S.
What is coplanar?A set of points in space are co planar, if there exists a geometric plane that contains them all.
What is non-collinear?Non-collinear points are points that do not lie on one straight line. The points are not collinear, when they are joined with each other, they form a triangle, which is a three-sided polygon.
For the given situation,
The diagram shows the plane M and the triangle.
In the diagram, all the points are lie on the same plane but the points are not lie on the same line.
Three points are always co planar, and if the points are distinct and non-collinear, the plane they determine is unique.
Hence we can conclude that option (C) non-collinear and co planar is the correct answer about the points R, L, and S.
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find the perimeter of the shaded figure.
Answers
Answer:
38
Step-by-step explanation:
Count the unit squares for each side, starting from the left it's 7, 10, 7, 2, 2, 6, 2, and 2
Now add them together:
7 + 10 + 7 + 2 + 2 + 6 + 2 + 2 = 17 + 9 + 8 + 4 = 26 + 12 = 38
Answer: 38 units
Step-by-step explanation: Perimeter is asking for the total length of all the sides that make up a figure. The top border is 10 units, the two outer sides are 7 each (14 in total), if you add all the downward facing edges you get another 10 units, and the two inside sides are 2 units each (4 in total).
10+14+10+4 = 38
Are these triangles similar? explain why did you pick yes or no.
Answers
14. The maximum capacity for seating in a theater is 500 people. The theater sells two types of
tickets, adult tickets for $7.25 each and child tickets for $4 each. If they sold out on a certain
showtime and made a total of $3,157 in ticket sales, how many of each type of ticket was sold for
that showtime?
Answers
Taking into account the definition of a system of linear equations, the amount of adult tickets sold is 356 and the amount of child tickets sold is 144.
System of linear equationsA system of linear equations is a set of two or more equations of the first degree, in which two or more unknowns are related.
Solving a system of equations consists of finding the value of each unknown so that all the equations of the system are satisfied. That is to say, the values of the unknowns must be sought, with which when replacing, they must give the solution proposed in both equations.
Amount of ticket soldIn this case, a system of linear equations must be proposed taking into account that:
"x" is the amount of adult tickets sold."y" is the amount of child tickets sold.The maximum capacity for seating in a theater is 500 people. If they sold out on a certain showtime, this is represented by x+y=500.
On the other side, the theater sells two types of tickets, adult tickets for $7.25 each and child tickets for $4 each and the made a total of $3,157. This is represented by 7.25x +4y=3157.
So, the system of equations to be solved is
\(\left \{ {{x+y=500} \atop {7.25x+4y=3157}} \right.\)
There are several methods to solve a system of equations, it is decided to solve it using the substitution method, which consists of clearing one of the two variables in one of the equations of the system and substituting its value in the other equation.
In this case, isolating the variable x from the first equation you get:
x=500 - y
Substituting this expression in the second equation you get:
7.25× (500 -y) +4y=3157
7.25×500 - 7.25y +4y=3157
3625 - 7.25y +4y=3157
- 7.25y +4y=3157 -3625
-3.25y= -468
y= (-468)÷ (-3.25)
y= 144
Substituting this value in the expression x=500 - y you get:
x=500 - 144
x=356
Remembering that "x" is the amount of adult tickets sold and "y" is the amount of child tickets sold, you get that the amount of adult tickets sold is 356 and the amount of child tickets sold is 144.
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Find the volume of the cone.
Use 3.14 for π. Round your answer to the nearest tenth if needed. Do not enter units.
Answers
\(\qquad\qquad\huge\underline{{\sf Answer}}♨\)
Volume of cone :
\(\qquad \sf \dashrightarrow \: \frac{1}{3} \pi{r}^{2} h\)
Now, let's plug in the values as per the diagram ~
\(\qquad \sf \dashrightarrow \: \frac{1}{3} \times 3.14 \times {(18)}^{2} \times 10\)
\(\qquad \sf \dashrightarrow \: \frac{1}{3} \times 31.4 \times 324\)
\(\qquad \sf \dashrightarrow \: 31.4 \times108\)
\(\qquad \sf \dashrightarrow \:3391.2 \: \: ft {}^{3} \)
Compare 4:10 and 12:20 using fractions. Which ratio is smaller? How do you know?
Answers
times the whole ratio by two to get: 8:20 now we can compare it to 12:20 and so 8 < 12 so the first one is smaller
Please help me find the area of this!!! 15 points!!
Answers
Answer:
\(A\approx1608.7\text{ yards}^2\)
Step-by-step explanation:
We can use the alternate formula for the area of a triangle:
\(A=\frac{1}{2}ab\sin(C)\)
Where a and b are the side lengths, and C is the angle between the side lengths.
We know that one side length is 60 and the other is 70.
The angle between them is 50°.
So, substitute 60 for a, 70 for b, and 50° for C. This yields:
\(A=\frac{1}{2}(60)(70)\sin(50)\)
Evaluate using a calculator. Make sure you’re in Degrees Mode.
Therefore, the approximate area is:
\(A\approx1608.7\text{ yards}^2\)
not sure on how to type this out but can anyone help me ??
Answers
Answer:
484
Step-by-step explanation:
This is because 22 squared is 484.
When you take something out from under the square root symbol, you have to square it.
Thus, a number outside the square root to the left of the symbol was originally that same number squared under the square root symbol.
Thus as 22 is outside the symbol, it was originally 22 squared while under the symbol, and 22 squared is 484.
484, because 22 squared is equivalent to 484z