ABC is a triangle in which angle B= 2 angle C. D is a point on BC such that AD bisects angle BAC and AB=CD. Prove that angle BAC=72°
Final answer:
Upon reviewing the proof and the assumed relationship between the angles, the given information leads to an incorrect conclusion of angle BAC being 90°. The mistake indicates a reassessment of angle relationships is required to determine the true measure of angle BAC in this question.
Explanation:
To prove that angle BAC is 72° in a triangle ABC where angle B is twice angle C and where AD bisects angle BAC with AB equal to CD, we proceed as follows:
Let angle BAC be represented as 2x. Therefore, since AD bisects angle BAC, each angle BAD and DAC is x.
Since angle B is twice angle C, let angle C be x and angle B then is 2x. It is given that AB is equal to CD, meaning triangle ABD is isosceles with angles BAD = DAC.
In isosceles triangle ABD, the angles at base AD are equal, which means each of these angles is x. Thus, the sum of angles in triangle ABD is x (at A) + 2x (at B) + x (at D) = 180°.
Combining these angles, we get 4x = 180°. Dividing both sides by 4, we obtain x = 45°.
Since angle BAC is 2x and x is 45°, angle BAC is therefore 90°.
This leads to a contradiction to the original assumption and upon review reveals the mistake in the assumption about the relationship of the angles given as twice. The correct relationship should be considered to find the accurate measure of angle BAC.
Clara writes the equation (x – 13)(x + 8) = 196 to solve for the missing side length of a triangle represented by the factor x + 8. What is the missing side length represented by x + 8 units of the triangle?
The price of a stock was $15.22 per share. an investor bought the most shares possible for $2000. to the nearest whole number, how many shares of stock did she buy?
a. 131
b. 133
c. 478
d. 761
Solve for x: -4 [x 5] = -16
The solution to the equation -4[x + 5] = -16 is x = -1.
To solve the equation -4[x + 5] = -16 for x, we will follow these steps:
First, we need to isolate the term with the variable. We can start by dividing both sides of the equation by -4 to get rid of the coefficient:
-4[x + 5] / -4 = -16 / -4
Simplifying both sides, we get:
x + 5 = 4
Next, solve for x by subtracting 5 from both sides:
x + 5 - 5 = 4 - 5
Simplifying this, we find:
x = -1
Consider the planes 2x + 1y + 5z = 1 and 2x + 5z = 0.
(A) Find the unique point P on the y-axis which is on both planes. ( ? , ? , ? )
(B) Find a unit vector u with positive first coordinate that is parallel to both planes.
? I + ? J + ? K
(C) Use the vectors found in parts (A) and (B) to find a vector equation for the line of intersection of the two planes,r(t) =
? I + ? J + ? K
To find the point P on the y-axis that is on both planes, set x = 0 and z = 0 for each plane and solve for y. The unique point P is (0, 1, 0). To find a unit vector u parallel to both planes, find the normal vectors of the planes and normalize one of them. The unit vector u with a positive first coordinate is (2/sqrt(30), 1/sqrt(30), 5/sqrt(30)). The vector equation for the line of intersection of the two planes is r(t) = (0, 1, 0) + t(5, -10, -2).
Explanation:To find the unique point P on the y-axis that is on both planes, we need to find the values of x, y, and z that satisfy the equations of both planes.
For the first plane, 2x + y + 5z = 1, if we set x = 0 and z = 0, we can solve for y:
0 + y + 0 = 1
y = 1
Therefore, the point P on the y-axis that is on the first plane is (0, 1, 0).
For the second plane, 2x + 5z = 0, since there is no y term, any point on the y-axis will satisfy this equation. So, we can choose any value for y and set x = 0 and z = 0. Let's choose y = 1:
2(0) + 5(0) = 0
Therefore, the point P on the y-axis that is on the second plane is (0, 1, 0).
Since both points have the same x and z coordinates, the unique point P on the y-axis that is on both planes is (0, 1, 0).
To find a unit vector u with a positive first coordinate that is parallel to both planes, we can find the normal vectors of the planes and then normalize one of them.
For the first plane, the normal vector is (2, 1, 5). For the second plane, the normal vector is (2, 0, 5). Both normal vectors are parallel to the planes.
Let's normalize the first vector:
||u|| = sqrt(2^2 + 1^2 + 5^2) = sqrt(30)
So, the unit vector u with a positive first coordinate that is parallel to both planes is (2/sqrt(30), 1/sqrt(30), 5/sqrt(30)).
To find a vector equation for the line of intersection of the two planes, we can take the cross product of the normal vectors of the planes to get a vector that is perpendicular to both planes.
Let's calculate the cross product:
(2, 1, 5) x (2, 0, 5) = (5, -10, -2)
Now, we can use one of the points on the line of intersection, which is (0, 1, 0), and the direction vector (5, -10, -2) to form the vector equation:
r(t) = (0, 1, 0) + t(5, -10, -2)
Find the slope and y-intercept of the line.
y = x – 5
A) slope: –5, y-intercept: 1
B) slope: –1, y-intercept: 5
C) slope: 5, y-intercept: –1
D) slope: 1, y-intercept: –5
Using the slope-intercept form, the slope and y-intercept of the line y = x – 5 is D) slope: 1, y-intercept: –5.
What is the slope?The slope is the ratio of the vertical changes to the horizontal changes between two points of the line.
The slope of the line is calculated as follows:
m = Δy/Δx
Given that the line y = x – 5
Let us consider
y = x – 5
Therefore, the slope of the line is;
m = 1
y-intercept of the line.
b = -5
Hence, the slope and y-intercept of the line y = x – 5 is D) slope: 1, y-intercept: –5
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Find the quadratic polynomial ax^2+bx+c which best fits the function f(x)=8^x at x=0, in the sense that g(0)=f(0), and f'(0)=g'(0), and f''(0)=g''(0).
g(x)=?
According to the synthetic division below, which of the following statements are true? Check all that apply.
_________
-7 ) 2 10 -20
-14 28
_____
2 -4 8
A. (x+7) is a factor of 2x^2+10x-20
B. (x-7) is a factor of 2x^2+10x-20
C. When x=7, 2x^2+10x-20=8
D. When x=-7, 2x^2+10x-20=8
E. When (2x^2+10x-20) is divided by (x+7), the remainder is 8
F. When (2x^2+10x-20) is divided by (x-7), the remainder is 8
Please help!
When [tex]x = - 7, 2{x^2} + 10x - 20 = 8[/tex] and If [tex]2{x^2} + 10x -20[/tex] is divided by [tex]\left( {x + 7} \right)[/tex], the remainder is 12. Option (D) is correct and option (E) is correct.
Further Explanation:
Explanation:
If division of a polynomial by a binomial result in a remainder of zero means that the binomial is a factor of polynomial.
The synthetic division can be expressed as follows,
[tex]\begin{aligned}- 7\left| \!{\nderline {\,{2\,\,\,\,\,\,\,\,\,\,10\,\,\,\,\,\,\,\,\,\, - 20} \,}} \right.\hfill\\\,\,\,\,\,\,\underline {\,\,\,\,\,\,\,\,\,\, - 14\,\,\,\,\,\,\,\,\,\,\,\,\,\,28} \hfill\\ \,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\, - 4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,8 \hfill\\\end{aligned}[/tex]
The last entry of the synthetic division tells us about remainder and the last entry of the synthetic division is [tex]8[/tex]. Therefore, the remainder of the synthetic division is [tex]8[/tex].
When [tex]x = - 7, 2{x^2} + 10x - 20 = 8[/tex] and If [tex]2{x^2} + 10x -20[/tex] is divided by [tex]\left( {x + 7} \right)[/tex], the remainder is 12. Option (D) is correct and option (E) is correct.
Option (A) is not correct.
Option (B) is not correct.
Option (C) is not correct.
Option (F) is not correct.
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Synthetic Division
Keywords: division, factor (x+7), remainder 8, statements, true, apply, divided, binomial synthetic division, long division method, coefficients, quotients, remainders, numerator, denominator, polynomial, zeros, degree.
Help! Match the term to the definition.
Perpendicular cross section of a pyramid
Perpendicular cross section of a cylinder
Parallel cross section of a sphere
Shape created when a rectangle is rotated about the y–axis
Shape created when a right triangle is rotated about the y–axis
A) rectangle
B) triangle
C) circle
D) cone
E) cylinder
Advance tickets for a school play went on sale. The price of each student ticket was $4 and everyone else paid $5. On the first day, no more than $80 in tickets were sold. Describe and explain the possible values of s, the number of student tickets sold, and e, the number of tickets sold to nonstudents.
Answer:
Partial and negative tickets cannot be sold, so the minimum number values of e and s are 0. If s = 0, then e = 16, and if e = 0, then s = 20. Therefore, the values of s are whole numbers from 0 to 20 and the values of e are whole numbers between 0 and 16. The greatest number of student tickets sold was 20 and the greatest number of nonstudent tickets sold was 16.
Step-by-step explanation:
write and equation for the line with a y-intercept of 5 that is perpendicular to the line with equation y=-3/4x+2
0.016km:8m:24m:= ----:----:21
The price of an item has been reduced by 85%. The original price was $80. What is the price of the item now?
The reduced price of the object is equal to $12.
What is the percentage?The percentage is defined as representing any number with respect to 100. It is denoted by the sign %. The percentage stands for "out of 100." Imagine any measurement or object being divided into 100 equal bits.
Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.
Given that the price of an item has been reduced by 85%. The original price was $80.
The reduced price of the item will be calculated as,
The reduction in price would be:
80 x 0.85 = 68
Price after discount is calculated as,
80 - 68 = $12
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Which is bigger, 0.25 or 0.6?
The yearbook club had a meeting. The meeting had 30 people, which is five-sixths of the club. How many people are in the club?
The student scores on mrs fredericks mathematics test are shown on the stem and leaf plot below
7 bicycles to 13 skateboards
Kim works as a salesperson for a photo studio, to find her earning for the week, she multiples his total sales by 0.175 her sales for the week is October 10 total 2,507.47 what did she earn for the week?
What is the length of the conjugate axis?
(x-1)^2/25-(y+3)^2/9=1 ...?
Answer:
Length of the conjugate axis is 6 units.
Step-by-step explanation:
Given Equation is ,
[tex]\frac{(x-1)^2}{25}-\frac{(y+3)^2}{9}=1[/tex]
To find: Length of the conjugate axis.
We know that given equation is Equation of Hyperbola.
First Transverse Axis: Axis passing through the vertices is called the transverse axis. Length of the transverse axis is 2a.
Now, Conjugate Axis: Axis which is perpendicular to the transverse axis through the center is called the conjugate axis. Length of the Conjugate axis is 2b.
Equation in standard form is written as ,
[tex]\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1[/tex]
So, Comparing with Standard equation,
we get, a² = 25 ⇒ a = 5 and b² = 9 ⇒ b = 3
Thus, Length of the conjugate axis = 2 × 3 = 6 unit
Therefore, Length of the conjugate axis is 6 units.
For the given equation of hyperbola:
[tex]\dfrac{(x-1)^2}{25}-\dfrac{(y+3)^2}{9}=1[/tex]
The length of the conjugate axis is 6 units.
In analytic geometry, a hyperbola is a conic section created when a plane meets a double right circular cone at an angle that overlaps both cone halves.
The standard equation of the hyperbola is,
[tex]\dfrac{(x-h)^2}{a^2}- \dfrac{(y-k)^2}{b^2} =1[/tex]
Where,
a is half of the transverse axis.
b is the half of the conjugate axis.
The given equation is,
[tex]\dfrac{(x-1)^2}{25}-\dfrac{(y+3)^2}{9}=1[/tex]
Comparing it with the above standard equation, we get:
a² = 25
⇒ a = ± 5
b² = 9
⇒ b = ± 3
Since, the length of the conjugate axis of hyperbola = 2b
Therefore,
The length of the conjugate axis:
= 2x3
= 6 units.
Hence,
The required length of the conjugate axis is 6 units.
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Suppose that F(x) = x^2 and G(x) = 2x^2-5. Which statement best compares the graph G(x) with the graph of F(x)?
A. The graph of G(x) is the graph of F(x) stretched vertically and shifted 5 units to the right
B. The graph of G(x) is the graph of F(x) compressed vertically and shifted 5 units down
C. The graph of G(x) is the graph of F(x) compressed vertically and shifted 5 units to the right
D. The graph of G(x) is the graph of F(x) stretched vertically and shifted 5 units down
Comparing the graphs of F(x) = [tex]x^2[/tex] and G(x) = [tex]2x^2[/tex] - 5 shows that G(x)'s graph is F(x)'s graph stretched vertically by a factor of 2 and then shifted 5 units down.
There is no horizontal shift involved.
Explanation:Comparing the functions F(x) = [tex]x^2[/tex] and G(x) = [tex]2x^2[/tex] - 5, we observe two main transformations applied to F(x) to obtain G(x).
First, the coefficient 2 in front of[tex]x^2[/tex] in G(x) indicates that the graph of F(x) is stretched vertically by a factor of 2.
This stretching makes the graph of G(x) stretch away from the x-axis, becoming narrower compared to F(x).
Second, the term -5 added to [tex]2x^2[/tex] suggests that the entire graph of F(x) after being stretched is then shifted 5 units down.
It's important to note that this vertical shift is down because of the negative sign in front of 5; there is no horizontal shift involved.
Therefore, the statement that best compares the graph of G(x) to the graph of F(x) is:
D. The graph of G(x) is the graph of F(x) stretched vertically and shifted 5 units down.Based on the analysis, the correct statement is:
D. The graph of G(x) is the graph of F(x) stretched vertically and shifted 5 units down.
1. Identify the base function:
Both F(x) = x^2 and G(x) = 2x^2-5 share the same base function, which is x^2. This means their graphs have the same basic shape, a parabola.
2. Analyze the transformations:
Vertical stretch: The coefficient of x^2 in G(x) is 2, which is a vertical stretch factor of 2 compared to F(x). This stretches the graph of G(x) vertically by a factor of 2, making it narrower.
Vertical shift: The constant term in G(x) is -5, which corresponds to a downward shift of 5 units compared to F(x). This moves the entire graph of G(x) 5 units down.
3. Combine the transformations:
The graph of G(x) is obtained by taking the graph of F(x), stretching it vertically by a factor of 2, and then shifting it down by 5 units.
For graph refer to image:
In expression 8x, what is 8 to x
What is the slope of the line whose equation is −48=2x−8y?
It has to be made into a fractionnnnn
A toy company manufactures arcade games. They are marketing a new pinball machine to children. It is similar in size to the adult version of the same game. Both the adult and child models are shown below: Adult pinball machine GAME with base ME measuring 35 inches and sides measuring 56 inches. Child pinball machine G prime A prime M prime E prime with base M prime E prime measuring 14 inches If the perimeter of the adult pinball machine is 167 inches, what is the length, in inches of Segment line G prime A prime? Type the numeric answer only in the box below.
The correct answer is 8
Answer:
8 inches
Step-by-step explanation:
Given,
In two quadrilateral GAME and G'A'M'E',
ME = 35 inches, AM = GE = 56 inches,
M'E' = 14 inches,
Also, the perimeter of quadrilateral GAME = 167 inches,
⇒ GA + AM + ME + GE = 167
⇒ GA + 56 + 35 + 56 = 167
⇒ GA + 147 = 167
⇒ GA = 20 inches.
Now, GAME is similar to G'A'M'E' are similar,
By the property of similar figures,
[tex]\frac{ME}{M'E'}=\frac{GA}{G'A'}[/tex]
[tex]\implies G'A'=\frac{M'E'\times GA}{ME}=\frac{14\times 20}{35}=\frac{280}{35}=8\text{ in}[/tex]
Hence, the length of Segment line G'A' is 8 inches.
Please Help,
Given z = 8 and x = 6, what is the ratio of x to z in the simplest form? ...?
The function h(x) is quadratic and h(3) = h(–10) = 0. Which could represent h(x)?
h(x) = x2 – 13x – 30
h(x) = x2 – 7x – 30
h(x) = 2x2 + 26x – 60
h(x) = 2x2 + 14x – 60
Answer:
h(x) = 2x2 + 14x – 60 (D)
Step-by-step explanation:
The quadratic function that represents h(x) with roots at 3 and -10 is h(x) = x^2 - 7x - 30 since it correctly reflects the sum and product of its roots.
The question asks us to find which function could represent h(x), given that h(3) = h(–10) = 0. This means the quadratic function has roots at x = 3 and x = -10. A quadratic function with these roots can be written as h(x) = a(x - 3)(x + 10), where a is a non-zero coefficient.
By expanding this expression, we would get h(x) = a(x^2 + 7x - 30). We can then see if any of the given options matches this form upon factoring out the leading coefficient.
Comparing the given options:
h(x) = x^2 - 13x - 30 does not match because the sum of the roots should be -7 based on the roots 3 and -10.h(x) = x^2 - 7x - 30 matches the expected roots since the sum of the roots (-7) and the product of the roots (-30) align with the roots 3 and -10.h(x) = 2x^2 + 26x - 60 does not match because the coefficient of x is positive, while the sum of the roots derived from the roots of the function is negative.h(x) = 2x^2 + 14x - 60 does not match for similar reasons to the previous one and also the coefficients do not align.Therefore, the correct function that could represent h(x) is h(x) = x^2 - 7x - 30.
What is the standard form of (4,4), perpendicular to y = -4/9x
Which system of equations models the problem if x represents the number of parrotfish Jenna bought and y represents the number of triggerfish she bought?
Jenna bought 280 tropical fish for a museum display. She bought 4 times as many triggerfish as parrotfish. How many of each type of fish did she buy?
A. {x+y=280
{y=4x
B.{x−y=280
{y=4x
C.{x+y=4
{y=280x
D.{x+4y=280
{y=4x
Answer:
The system of equations is the option A
[tex]x+y=280[/tex]
[tex]y=4x[/tex]
Jenna bought [tex]56[/tex] parrot fish and [tex]224[/tex] trigger fish
Step-by-step explanation:
Let
x-----> the number of parrot fish
y-----> the number of trigger fish
we know that
[tex]x+y=280[/tex] -----> equation A
[tex]y=4x[/tex] ------> equation B
The answer Part 1) is the option A
Substitute equation B in equation A
[tex]x+4x=280[/tex]
Solve for x
[tex]5x=280[/tex]
[tex]x=56[/tex]
Find the value of y
[tex]y=4x[/tex]
[tex]y=4*56=224[/tex]
therefore
The answer part 2) is
Jenna bought [tex]56[/tex] parrot fish and [tex]224[/tex] trigger fish
Final answer:
The correct system of equations modeling Jenna's purchase of tropical fish, where x is the number of parrotfish and y is the number of triggerfish, is x + y = 280 and y = 4x, which is option A.
Explanation:
The problem given is about modeling a real-world situation with a system of equations where x represents the number of parrotfish Jenna bought and y represents the number of triggerfish she bought. Jenna bought 280 tropical fish for a museum display, and she bought 4 times as many triggerfish as parrotfish. The system of equations that models this situation is:
x + y = 280
y = 4x
This matches option A from the given choices. The first equation represents the total number of fish Jenna bought, while the second equation represents the relationship between the number of triggerfish and parrotfish, with the triggerfish being four times the number of parrotfish. To solve, substitute the second equation into the first to find the values of x and y.
Write a paragraph proof.
Given: line BC is congruent to line EC and line AC is congruent to line ED
Prove: line BA is congruent to line ED
Answer:
see the explanation
Step-by-step explanation:
we know that
The Side Angle Side postulate (SAS) states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then these two triangles are congruent.
In this problem
Triangles ABC and DEC are congruent by SAS postulate
Because
BC≅EC
AC≅DC
and the include angle
m∠BCA≅m∠ECD ----> by vertical angles
Remember that
If two triangles are congruent, its corresponding sides and corresponding angles are congruent
therefore
BC≅EC
AC≅DC
and
BA≅ED
During the revising stage in the writing process, the author _____.
Solve the following equation.-3 = x/5