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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.

Using the slope-intercept form, the slope and y-intercept of the line y = x – 5 is D) slope: 1, y-intercept: –5.

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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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).

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)

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

The average mileage covered per day is (x + y + z) / 3. It provides a balanced representation of the man's daily driving performance throughout the three-day period.

To find the average mileage covered per day, you need to calculate the total mileage covered over the three days and then divide it by the number of days (which is 3 in this case).

The total mileage covered over the three days is: x + y + z

The average mileage covered per day is: (x + y + z) / 3

This formula finds the mean distance covered each day.

By dividing the total distance by the number of days, the average mileage smooths out any fluctuations in daily distances and gives a more comprehensive view of his overall performance.

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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.

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:

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:

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.

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:

The reduced price of the object is equal to $12.

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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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.