Which best explains whether or not all isosceles triangles are similar?

All isosceles triangles are similar. Two angles within each triangle are always congruent.
All isosceles triangles are similar. The triangle sum theorem states that the sum of the angles in a triangle is 180°. Therefore, the third angle can always be determined.
All isosceles triangles are not similar. The pair of congruent angles within one triangle is not necessarily congruent to the pair of congruent angles within the other triangle.
All isosceles triangles are not similar. Given only the vertex angle of an isosceles triangle, there is not enough information to determine the measures of the base angles. Therefore, it is not possible to determine if the base angles of one isosceles triangle are congruent to the base angles of another.

The Law of Cosines is used to find the remaining parts of a triangle when the triangle is not a right triangle and when either the lengths of two sides and the measure of the included angle is known (SAS) or the lengths of the three sides (SSS) are known.

The law of cosine states if a,b,c are the three sides of any triangle then [tex] a^{2} =b^{2} +c^{2} -2bccosA [/tex]

In the above proof in statement 8 the x in statement 5 is replaced by c cos A from statement 6.

Among the given options ,option C holds true as the reason for statement 8.

Answer:

72 is the og

Step-by-step explanation:

72*.17=12.24

The original cost of the meal is $72.

In mathematics →

Given is that Sally left a $12.24 tip that represented 17% of her bill.

Assume that the original cost of the meal is $[x]. So, we can write -

12.24 = 17% of {x}

12.24 = (17/100) × {x}

{x} = (12.24 x 100)/17

{x} = 1224/17

{x} = 72

Therefore, the original cost of the meal eaten by Sally is $72.

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Answer:

C : angle B is 72.7°

Step-by-step explanation:

I did the same Assingnment

Answer:

49 odd numbers are there between 101 and 199.

Step-by-step explanation:

Given  : 101 and 199.

To find : How many odd integers are there between 101 and 199.

Solution : We have given  101 and 199.

Total number(N) between 101 and 199

N = 199 - 101 = 98

N = 98 .

We can see 98 is even number so, there are half numbers are odd and half numbers are even .

Odd number = [tex]\frac{N}{2}[/tex].

Odd number =  [tex]\frac{98}{2}[/tex].

Odd number = 49.

Therefore, 49 odd numbers are there between 101 and 199.

Final answer:

The probability that a DVD will have a video or audio defect is 1.9%, calculated using the principle of inclusion-exclusion. The probability that a DVD will not have any defect is 98.1%, obtained by subtracting the probability of having a defect from 100%.

Explanation:

To solve the probability that a DVD purchased by a customer will have a video or audio defect or not, we can use the principle of inclusion-exclusion for probabilities.

According to the given data:

Part a: The probability that a DVD will have a video or audio defect (at least one type of defect) is calculated by adding the probabilities of each defect and subtracting the probability of both defects occurring, as these are counted twice.

P(either type of defect) = P(video defect) + P(audio defect) - P(both defects)

P(either type of defect) = 1.5% + 0.8% - 0.4% = 1.9%

Part b: To find the probability that a DVD will not have any defect, we subtract the probability of either defect from 100%.

P(no defect) = 100% - P(either type of defect)

P(no defect) = 100% - 1.9% = 98.1%

Therefore, the probability that a DVD will have a video or audio defect is 1.9%, and the probability that a DVD will not have any defect is 98.1%.

Answer:

Its B

Step-by-step explanation:

I just took the test

Final answer:

The trigonometric identity (tan(x) + tan(y)) / (1 - tan(x)tan(y)) = (sin(x)cos(y) + cos(x)sin(y)) / (cos(x)cos(y) - sin(x)sin(y)) can be verified by using the angle addition formulas for sine and cosine, and by applying the tangent sum formula. Both sides reduce to tan(x + y), confirming the identity.

Explanation:

The given trigonometric identity to verify is:
(tan(x) + tan(y)) / (1 - tan(x)tan(y)) = (sin(x)cos(y) + cos(x)sin(y)) / (cos(x)cos(y) - sin(x)sin(y)).

Trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables where both sides of the equation are defined.

Let's verify the given identity step by step. First, rewrite the right-hand side using the angle addition formulas for sine and cosine:

So, the right-hand side can be written as sin(x + y) / cos(x + y). This is equivalent to tan(x + y).

Now, let's examine the left-hand side and apply the tangent sum formula:

As we can see, the left-hand side is exactly the tangent sum formula; therefore it also simplifies to tan(x + y).

Since tan(x + y) is a common form for both sides of the identity, we can conclude that the identity is verified:

(tan(x) + tan(y)) / (1 - tan(x)tan(y)) = sin(x + y) / cos(x + y)

Final answer:

The sum of the series 4+12+36+108+...+8748 is found by observing that it is a geometric progression with a common ratio of 3. By comparing the last term to the options, and understanding the rapidly increasing nature of geometric series, the sum is slightly larger than the last term, which only matches the option 8908.

Explanation:

To find the sum of the geometric series 4+12+36+108+...+8748, we first need to identify the common ratio (r) of the series. Each term is obtained by multiplying the previous term by the constant r. In this case, r = 12 / 4 = 36 / 12 = 108 / 36 = 3. The series is therefore a geometric progression with the first term a = 4 and common ratio r = 3.

We can then use the formula for the sum of the first n terms of a geometric series, which is Sn = a(1-rn) / (1-r), where r ≠ 1. However, we need the number of terms (n). To find n, we use the formula for the nth term of a geometric sequence, which is an = a * rn-1. We can solve for n considering the last term given: 8748 = 4 * 3n-1.

However, instead of manually finding the value of n and then the sum, let us approach the problem with the available options. The sum of a geometric series increases very rapidly due to the exponential nature of each subsequent term. Comparing the last term with the provided options, we can infer that the sum has to be a little bit greater than the last term of the series, 8748.

By inspection of the options, the only number that is slightly greater than 8748 is 8908. Therefore, to the quickest approximation and without going through detailed calculations, the sum of the series is 8908.

the answer is $60.30 because half of 100.50 is 50.25 and the both would pay 10%. 10% of 100.50 is 10.05.

10.05+50.25= 60.30

In the question it is already given that Melissa and Joey would have to share the total bill along with the tips at the resturant during their dinner. The paid the water an amount of $100.50 and additional 20% tip.Firstly we have to find the amount of the tip and then add with the total amount to calculate the total amount paid by both of them.

Amount of tips paid to the waiter = (20/100) * 100.50 dollars

                                                    = 2010/100 dollars

                                                    = 20.10 dollars

So Melissa and Joey paid $20.10 as tips to the waiter.

Then the total amount paid  by both of them = (100.50 + 20.10) dollars

                                                                      = 120.60 dollars

Then the amount paid by Melissa and Joey = 120.60/2 dollars

                                                                    = 60.3 dollars.

So Melissa and Joey will individually pay $60.3 each.

Answer: decreases by 3%

Step-by-step explanation:

Answer:  The vertex form of the given function is [tex]y=5\left(x+\dfrac{3}{2}\right)^2-\dfrac{53}{4}[/tex] and its axis of symmetry is [tex]x=-\dfrac{3}{2}.[/tex]

Step-by-step explanation:  The given quadratic function is

[tex]y=5x^2+15x-2~~~~~~~~~~~~~~(i)[/tex]

We are to rewrite the above function in vertex form and to determine its axis of symmetry.

We have from equation (i),

[tex]y=5x^2+15x-2\\\\\Rightarrow y=5(x^2+3x)-2\\\\\Rightarrow y=5\left(x^2+2\times x\times \dfrac{3}{2}+\dfrac{9}{4}\right)-\dfrac{45}{4}-2\\\\\Rightarrow y=5\left(x+\dfrac{3}{2}\right)^2-\dfrac{45+8}{4}\\\\\Rightarrow y=5\left(x+\dfrac{3}{2}\right)^2-\dfrac{53}{4}.[/tex]

So, the given function is a parabola with vertex at the point [tex]\left(-\dfrac{3}{2},-\dfrac{53}{4}\right).[/tex]

Therefore, the axis of symmetry is given by

[tex]x=-\dfrac{3}{2}.[/tex]

Thus, the vertex form of the given function is [tex]y=5\left(x+\dfrac{3}{2}\right)^2-\dfrac{53}{4}[/tex] and its axis of symmetry is [tex]x=-\dfrac{3}{2}.[/tex]

To convert the quadratic function y = 5x² + 15x - 2 into vertex form, completing the square yields y = 5(x + 1.5)² - 13.25, with an axis of symmetry at x = -1.5.

To rewrite the quadratic function y = 5x² + 15x - 2 in vertex form and find the axis of symmetry, follow these steps:

Complete the square

Start with the given quadratic function:

y = 5x² + 15x - 2

To complete the square, first factor out the coefficient of x² (which is 5):

y = 5(x² + 3x) - 2

Now, focus on the expression inside the parentheses, x² + 3x.

To complete the square, need to add and subtract a constant term that will make it a perfect square trinomial.

add (3/2)² = 2.25 and subtract the same value inside the parentheses:

y = 5(x² + 3x + 2.25 - 2.25) - 2

Simplify the expression inside the parentheses

Now, simplify the expression inside the parentheses:

x² + 3x + 2.25 = (x + 1.5)²

So, the equation becomes:

y = 5[(x + 1.5)² - 2.25] - 2

Distribute the 5 and simplify further

y = 5(x + 1.5)² - 5(2.25) - 2

Rewrite in vertex form

y = 5(x + 1.5)² - 11.25 - 2

y = 5(x + 1.5)² - 13.25

Now, the quadratic function is in vertex form:

y = 5(x + 1.5)² - 13.25

Find the axis of symmetry

The axis of symmetry of a parabolic equation in the form y = a(x - h)² + k is given by x = h.

The axis of symmetry is x = -1.5 (opposite sign of the value inside the parentheses).

Therefore, the axis of symmetry for the quadratic function y = 5x² + 15x - 2 is x = -1.5.

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