Is it possible to have 3 acute angles in a triangle?

We prooved the paragraph proof of Theorem 3.8 that if two lines are perpendicular to the same line, then they are parallel to each other.

In the given question we have to explain the a paragraph proof of Theorem 3.8.

Let the lines that cross line a be m and n. Let m be parallel to a. As a result, the intersection of m and a produces four 90° angles.

Let n be parallel to a. This indicates that each of the four angles created by the intersection of n and an is 90 degrees.

The same-side internal angles (between lines m and n) are also congruent because all of the angles are congruent. The lines are parallel if two internal angles on the same side are congruent.

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The right question is:

Write a paragraph proof of theorem 3-8: in a plane, if two lines are perpendicular to the same line, then they are parallel to each other.

The transformed equation of  is  y = x³ is y =-1/7(x+8)³

Given that :

The equation of y=x^3 and Vertical compression by a factor of 1/7 ,horizontal shift 8 units to the left, reflection across the x axis

Applying the given transformations, we have:

To have a vertical compression by a factor of 1/7, we need to multiply the function by 1/7. So, we have:

y=1/7x³

We must add 8 to x in order to have a horizontal shift of 8 units to the left. We thus have:

y=1/7(x+8)³

A reflection over the x-axis would be the final step, we need to multiply the function by −1. So, we have:

y= -1/7(x+8)³

Therefore, the transformed equation is y= -1/7(x+8)³

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17 and 19 are two consecutive odd integers such that the square of the lesser integer increased by six times the greater integer is 403

A quadratic equation is a second-order polynomial equation in a single variable x , ax2+bx+c=0. with a ≠ 0 .

Consecutive odd integers differ by 2  (1, 3, 5, 7, etc.) so if your first integer is n, your second and consecutive odd integer would be (n + 2).

The question says that  the square of the lesser integer increased by six times the greater integer is 403

n²+6(n+2)=403

n²+6n+12=403

n²+6n-391=0

Use quadratic formula

n=-b±√b²-4ac/2a

=-6±√6²-4(-391)/2

n=17 and n=-23

Hence, the two consecutive integers be 17 and 19.

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Answer:

A trapezoid is isosceles if and only if its diagonals are congruent. So if we can prove that the bases are parallel and the diagonals are congruent, then we know the quadrilateral is an isosceles trapezoid, as Cool Math accurately states.

Step-by-step explanation:

Slope-intercept form, the equation is given by  y = -7.

What is Slope?

A line's steepness can be determined by looking at its slope. Slope is calculated mathematically as "rise over run" (change in y divided by change in x).

What is Slope-intercept?

When you know the slope of the line to be investigated and the given point is also the y intercept, you can utilize the slope intercept formula, y = mx + b. (0, b). The y value of the y intercept point is denoted by the symbol b in the formula.

Since x=13 is a vertical line and the line must be perpendicular to it, it must be horizontal, or have a slope of zero.

The equation of the line can be expressed using the point-slope form as

or   [tex]y-(-7)=0(x-1)[/tex]

or  [tex]y + 7 = 0[/tex]

In slope-intercept form, the equation is given by

              y = -7

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The number of days Alyssa can afford to rent can be represented with the inequality x ≤ 5.2

A rental car company charges $37.50 per day to rent a car and $0.05 for every mile driven.

Alyssa wants to rent a car, knowing that:

Therefore, the inequality which can be used to determine x the number of days Alyssa can afford to rent  while staying within her budget can be calculated as follows;

x = number of days she can rent

37.50x + 0.05(100) ≤ 200

37.50x + 5 ≤ 200

37.50x  ≤ 195

divide both sides by 37.50

x ≤ 195 / 37.50

x ≤ 5.2

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Using the slope formula, the lines are:

1. Neither

2. Perpendicular

3. Neither

The slope formula is given as, m = change in y / change in x.

Slope of lines that are parallel are the same while slope of perpendicular line will give a product of -1.

1. Slope of L1 (m) = change in y / change in x = (6 - 0) / (0 - (-2))

m = 6/2

m = 3

Slope of L2 (m) = change in y / change in x = (3 - 3) / (0 - 1)

m = 0/-1

m = 0

Both lines are neither.

2. Slope of L1 (m) = change in y / change in x = (-2 - 2) / (8 - 6)

m = -4/2

m = -2

Slope of L2 (m) = change in y / change in x = (0 - 1) / (3 - 5)

m = -1/-2

m = 1/2

Both lines are perpendicular because (-2)(1/2) = -1

3. Slope of L1 (m) = change in y / change in x = (4 - (-1)) / (4 - (-5))

m = 5/9

Slope of L2 (m) = change in y / change in x = (10 - (-7)) / (8 - 4)

m = 17/4

The lines are neither.

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Answer:

2x - 9

Step-by-step explanation:

x = unknown number

2 * x -9

2x - 9

Yes, a cubic graph is an exponential.

What is Cubic Graph?

A cubic graph in the study of graph theory is a graph with all of its vertices having degree three. A cubic graph is a 3-regular graph, to put it another way. Trivalent graphs are another name for cubic graphs. A cubic bipartite graph is known as a bicubic graph.

What is Exponential?

The mathematical expression f(x)=exp or ex denotes the exponential function. Unless otherwise stated, the phrase usually refers to the positive-valued function of a real variable, though it can also be applied to complex numbers or extrapolated to other mathematical objects like matrices or Lie algebras.

Graphs of Exponential Functions if;

When the base is greater than 1, the exponential function has the following characteristics and you can easly understand from this :

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The area of the path is 456 square feet

The formula for determining area of a rectangle is given as;

A = lw

Where;

Now, substitute the values into the formula, we have;

Area of the rectangle  = 40 × 70

Area of the rectangle = 2800 square feet    (1)

Bordered fence with a 4 ft path so dimensions 44 ft by 74 ft

Area of rectangular garden with Path = 44 × 74

= 3256 square feet       (2)

Area of the walk = (2) - (1) = 3256 - 2800 = 456 square feet

Hence, the value is 456 square feet

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Answer:

expanded: 7+0.08

standard: 7.08

Step-by-step explanation:

The hundredths place is the second digit after the decimal, so 8 hundredths is 0.08.

Answer:

The integers could be 8 and 4, just checked with a calculator

Based on the concept of ASA, when two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

Congruent triangle:

Congruent triangle means all three corresponding sides are equal and all the three corresponding angles are equal in measure.

Given,

Here we need to prove that triangle congruence by ASA.

In order to prove this one, we have to know that definition of ASA,

ASA also known as Angle-side-angle is a rule used to prove whether a given set of triangles are congruent.

In geometry, ASA rule states that "If two angles and the included side of one triangle are equal to two angles and included side of another triangle, then the triangles are congruent."

When we compare the given triangle to this rule then we clearly identify whether the triangle is congruent or not.

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Answer:

add up the exponents of each term and select the highest sum.

The number of nickels and the number of dimes that Joseph has are 12 and 5 respectively.

An equation of degree one is known as a linear equation.

A linear equation of two variables can be represented by ax + by = c.

Let the no. of nickels be N and no. of dimes be D.

∴ N + D = 17...(i) and 0.05N + 0.1D = 1.45...(ii)

Multiplying (ii) by 100 we get 5N + 10D = 145...(iii)

Now multiply (i) by 5 we get 5N + 5D = 85...(iv)

Now subtracting (iv) from (iii) we get,

5D = 60.

D = 12 hence N = 5.

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The true statements are

The side length of the transparent cube is given as

Length = 10 millimetres

The volume of the cube is calculated as

Volume = Length³

Substitute the known values in the above equation, so, we have the following representation

Volume =10³

Evaluate

Volume = 1000 cubic millimetres

1000 cubic millimetres and 30 cubic millimetres do not have the same value

So, the co-worker is incorrect.

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Answer:

Step-by-step explanation:

You want a transformation that maps figure P in the 3rd quadrant to figure Q in the second quadrant.

The figures have the same orientation (sequence of angles and side lengths), so reflection is not needed as part of the transformation.

The shorter mid-length side in Figure P has a slope of -1, and the corresponding side in Figure Q has a slope of +1. This can be the result of a 90° clockwise rotation about the origin.

Rotation 90° CW about the origin will map the right-angle corner from (-7, -7) to (-7, 7).

The right-angle corner in Figure Q is at (-7, 12), so a translation upward after the rotation is required. The amount of that translation is 12-7 = 5 units.

Translation upward by 5 units will map the right angle corner from its rotated position at (-7, 7) to its position in Figure Q at (-7, 12).

The series of transformations could be ...

__

Additional comment

The required transformation can be accomplished in one step by rotation 90° CW about the point (2.5, 2.5).

If Figure P is rotated 90° CW about its right-angle corner, then translation up by 19 units will map it to Figure Q. That is, there are many combinations of rotation and translation that will do the job.

1 must be the root of the polynomial [tex]$3 x^2+x-4$[/tex].

We are given that polynomial [tex]$3 x^2+x-4$[/tex] is being divided by factor [tex]$(\mathrm{x}-1)$[/tex]

It is also said that when David perform the mathematical operation it gives remainder 0 .

Remainder 0 represents that divisor [tex]$(\mathrm{x}-1)$[/tex] is a factor polynomial [tex]$3 x^2+x-4$[/tex].

As x-1 is a factor of the given ploynomial, we can get one of the root of by setting it equal to 0 and solve for x.

Therefore,

x-1=0

Adding 1 on both sides, we get

[tex]$$\begin{aligned}& x-1+1=0+1 \\& x=1 .\end{aligned}$$[/tex]

Therefore, 1 must be the root of the polynomial [tex]$3 x^2+x-4$[/tex].

Polynomials are sums of terms of the form k⋅xⁿ, where k is any number and n is a positive integer. For example, 3x+2x-5 is a polynomial. Introduction to polynomials. This video covers common terminology like terms, degree, standard form, monomial, binomial and trinomial.

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Option B, The table does not show a probability distribution.  the sum of all the probabilities is not equal to 1 .

A statistical function called a probability distribution explains the range of values and probabilities that a random variable might have. A statistical function known as a probability distribution explains the possibility of obtaining each potential value that a random variable can have.

The sum of the values is given as 0.001 + 0.008 + 0.034 + 0.057

= 0.1

Due to the fact that this is not a probability distribution.

We must choose the standard deviation as well. There is no probability distribution displayed in the table. These exist because sum = 1, which is not present here, is required for a probability distribution.

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Answer:

0.80

Step-by-step explanation:

The hundredth is 2 digits after the decimal.

Rounding 0.799, we want 0.xx

Look at the third digit, 9, meaning we round up.

However, since the second digit after the decimal is also a 9, we can make that a 0 by adding 1 to the first digit after the decimal.

We get 0.80

The inverse function of the function f(x) = 2x - 4 will be x/2 +2.

What is inverse function?

A function that returns the initial value for which a function has produced output is called an inverse function. If f(x) is a function that produces the value y, then f-1(y), which is the inverse function of y, will produce the value x. Inverse and reciprocal functions should not be confused. The output's original value is returned by the function's inverse, which is represented by the symbol f-1 (x). While the reciprocal of a function is represented by 1/f(x) or f(x)-1

Those that do are referred to as invertible. For a function f: X Y to have an inverse, it must possess the characteristic that there is exactly one x in X such that f(x) = y for every y in Y. This characteristic guarantees the existence of a function g: Y X that has the required connection to f.

f(x) = 2x - 4

y = 2x - 4

So x = y +4 /2

x = y/2 +2

Now to get the inverse of the function  put the x value in y

So [tex]f^{-1}[/tex](x) = x/2 +2

Hence, inverse function will be x/2 +2.

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Option C is correct that is the point of the triangle's angle bisectors where is the center of the largest circle that we could draw inside a given triangle.

Given that,

We have to find what is the center of the largest circle that we could draw inside a given triangle.

The options are

A. the triangle's medians' point of convergence.

B. the point where the triangle's sides' perpendicular bisectors converge.

C. the intersection of the triangle's angle bisectors.

D. the triangle's point of congruence for the triangle's heights

We know that,

An inscribed circle is the biggest circle that can be drawn inside of a triangle. The incenter is the name given to this circle's center.

All three of the triangle's angles' angle bisectors come together to form the incenter.

Therefore, Option C is correct that is the intersection of the triangle's angle bisectors where is the center of the largest circle that we could draw inside a given triangle.

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No, An arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant.

What is Arithmetic sequence ?

An arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant.

Example

2,4,6,8,10….is an arithmetic sequence with the common difference 2.

If the first term of an arithmetic sequence is a1 and the common difference is d, then the nth term of the sequence is given by:

aₙ=a1+(n−1)d

An arithmetic series is the sum of an arithmetic sequence. We find the sum by adding the first, a1 and last term, an, divide by 2 in order to get the mean of the two values and then multiply by the number of values, n:

Sₙ= [tex]\frac{n}{2}[/tex](a1+aₙ)

Example

Find the sum of the following arithmetic series 1,2,3…..99,100

We have a total of 100 values, hence n=100. Our first value is 1 and our last is 100. We plug these values into our formula and get:

S₁₀₀ = [tex]\frac{100}{2} (1+100)[/tex] =5050

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Answer:

change 5 weeks to days(5*7)-35days

then convert the days to seconds

35*86400/10*86400

answer 7/2

Answer: y=x−2 y = x − 2

Step-by-step explanation:

Answer:

4 [tex]\frac{1}{8}[/tex]

Step-by-step explanation:

Answer:

No.

Step-by-step explanation:

Because the equation has an exponent, it is quadratic, and not linear.

Answer:1

Step-by-step explanation:

Suppose that IQ scores in one region are normally distributed with a standard deviation of 14.  the mean IQ score for this region is 97.2.

z =(x-mean)/standard deviation

58th percentile is a z-score of 0.202

Hence,

0.202=(100-mean)/14

2.828=100-mean

Multiplying through by 14

Mean IQ score = 100- 2.828

Mean IQ score =97.17

Mean IQ score =97.2 (Approximately)

Therefore the mean IQ score for this region is 97.2 .

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The total number of cartons needed more to pack small 15 cartons than 35 large cartons is108 cartons.

Division is defined as one of the four basic operations of arithmetic in mathematics.

The parameters for this question are:

Total number of cartons =2835

Number of small cartons needed = 15 cartons

Number of large cartons needed = 35 cartons

To package the orange juice in large cartons we need [tex]\frac{2835}{35} =81 cartons[/tex]

Also, to package 15 small cartons we need [tex]\frac{2835}{15} =189 cartons[/tex]

Number of more small cartons than large cartons is

Small cartons -large cartons

=189-81=108 more cartons.

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Applying the definition of an Isosceles triangle, given that isosceles triangle TIC has a vertex angle I of 90 degrees, the measure of angle T is 45 degrees.

In geometry, an isosceles triangle is a triangle that has at least two sides of equal length. Sometimes it is specified as having exactly two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case. Examples of isosceles triangles include the isosceles right triangle, the golden triangle, and the faces of bipyramids and certain Catalan solids.

The two equal sides are called the legs and the third side is called the base of the triangle. The other dimensions of the triangle, such as its height, area, and perimeter, can be calculated by simple formulas from the lengths of the legs and base. Every isosceles triangle has an axis of symmetry along the perpendicular bisector of its base. The two angles opposite the legs are equal and are always acute, so the classification of the triangle as acute, right, or obtuse depends only on the angle between its two legs.

An Isosceles triangle has a vertex angle and two equal base angles.

Given that triangle TIC is an isosceles triangle:

angle I is the vertex angle = 90 degrees

angle T is a base angle

angle C is also a base angle

Thus:

angle C and angle T are congruent or equal to each other.

Therefore,

    Angle T = (180 - 90)/2

⇒ Angle T = 90/2

⇒ Angle T = 45 degrees.

applying the definition of an isosceles triangle, given that isosceles triangle TIC has a vertex angle I of 90 degrees, the measure of angle T is 45 degrees.

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