For the following figure, complete the statement for the specified points.
Points A, B, C, and Q are _____.

collinear
coplanar
both collinear and coplanar
neither collinear nor coplanar

Answer:

  2√42 ≈ 12.961

Step-by-step explanation:

All of the right triangles are similar, so ...

  short leg/long leg = BC/AC = AC/CD

  AC² = BC·CD = 7·24 = 168

  AC = √168 = 2√42

we know that

The Heron's Formula is a method for calculating the area of a triangle when you know the lengths of all three sides. The formula is equal to

[tex]Area=\sqrt{p(p-a)(p-b)(p-c)}[/tex]

where

a,b,c -----> are the lengths of the sides of a triangle

p ----> is half the perimeter

we have

[tex]J(-2,1)\ K(4,3)\ L(-2,-5)[/tex]

The formula to calculate the distance between two points is equal to

[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]

Step 1

Find the distance JK

[tex]J(-2,1)\ K(4,3)[/tex]

Substitute the values in the formula of distance

[tex]d=\sqrt{(3-1)^{2}+(4+2)^{2}}[/tex]

[tex]d=\sqrt{(2)^{2}+(6)^{2}}[/tex]

[tex]dJK=\sqrt{40}\ units[/tex]

Step 2

Find the distance KL

[tex]K(4,3)\ L(-2,-5)[/tex]

Substitute the values in the formula of distance

[tex]d=\sqrt{(-5-3)^{2}+(-2-4)^{2}}[/tex]

[tex]d=\sqrt{(-8)^{2}+(-6)^{2}}[/tex]

[tex]dKL=10\ units[/tex]

Step 3

Find the distance JL

[tex]J(-2,1)\ L(-2,-5)[/tex]

Substitute the values in the formula of distance

[tex]d=\sqrt{(-5-1)^{2}+(-2+2)^{2}}[/tex]

[tex]d=\sqrt{(-6)^{2}+(0)^{2}}[/tex]

[tex]dJL=6\ units[/tex]

Step 4

Find the perimeter of the triangle

we know that

the perimeter of a triangle is the sum of the length sides of the triangle

so

[tex]P=dJK+dKL+dJL[/tex]

substitute the values

[tex]P=\sqrt{40}\ units+10\ units+6\ units=22.32\ units[/tex]

Find half the perimeter

[tex]p=22.32/2=11.16\ units[/tex]

Step 5

Find the area of the triangle

Applying the Heron's Formula

[tex]Area=\sqrt{p(p-a)(p-b)(p-c)}[/tex]

we have

[tex]p=11.16\ units[/tex]

[tex]a=dJK=\sqrt{40}\ units=6.32\ units[/tex]

[tex]b=dKL=10\ units[/tex]

[tex]c=dJL=6\ units[/tex]

substitute the values

[tex]Area=\sqrt{11.16(11.16-6.32)(11.16-10)(11.16-6)}[/tex]

[tex]Area=\sqrt{11.16(4.84)(1.16)(5.16)}[/tex]

[tex]Area=17.98\ units^{2}[/tex]

[tex]Area=18\ units^{2}[/tex]

therefore

the answer is

The area of the triangle is [tex]18\ units^{2}[/tex]

The area of the triangle with given vertices is 18 units squared.

To find the area of a triangle with given vertices, we can use the formula:

Area =  1/2 * |(x1 * (y2 - y3) + x2 * (y3 - y1) + x3 * (y1 - y2))|

Plugging in the coordinates of the vertices:

A = 1/2 * |(-2 * (3 - (-5)) + 4 * (-5 - 1) + (-2) * (1 - 3))|

Simplifying the expression:

A = 1/2 * |(-2 * 8 + 4 * (-6) + (-2) * (-2))|

A = 1/2 * |(-16 - 24 + 4)|

A = 1/2 * |-36|

A = 1/2 * 36

A = 18

Therefore, the area of the triangle is 18 units squared.

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To evaluate tan(sin(90°)), first determine sin(90°) which equals 1. However, tan(1) is not a standard unit circle value, and if the question intended to find tan(90°), this would be undefined.

The correct option is (d).

The question is asking to use the unit circle to find the value of tan(sin(90°)).

Firstly, the sin(90°) on the unit circle is 1, as the sine function corresponds to the y-coordinate of the unit circle at that angle. Therefore, sin(90°) = 1.

Now, we need to find the tangent of this value, which is tan(1). However, tan(1) is not solvable by standard unit circle values, so it seems we may have a misunderstanding in the question. If the intention was to find tan(90°) instead, then the answer would be undefined as the tangent function at 90 degrees is undefined due to division by zero.

the average rate of change for the given graph from x = -2 to x = 0  is:

                                    3

We know that the average rate of the graph of the function f(x) form x=a to x=b is calculated by the formula:

[tex]Average\ Rate\ of\ change=\dfrac{f(b)-f(a)}{b-a}[/tex]

Here we have:

a= -2 and  b=0

Also, from the graph of the function we have:

f(a)=0  and  f(b)=6

Hence, the average rate of change of the graph from x= -2 to x=0 is:

[tex]Average\ Rate\ of\ change=\dfrac{6-0}{0-(-2)}\\\\i.e.\\\\Average\ Rate\ of\ change=\dfrac{6}{2}\\\\i.e.\\\\Average\ Rate\ of\ change=3[/tex]

              Hence, the answer is:  3

this is the question...

All the three figures given in the problem cannot be squares.

A parallelogram is a type of quadrilateral of which the opposite sides are equal and parallel. A diagonal of a parallelogram divides it into two congruent triangles.

Given that,

There are an equiangular rhombus, an equilateral rectangle, and a parallelogram with four right angles.

They can be explained one by one as follows,

(a) A rhombus has all the four sides as equal and its opposite sides are parallel.

When all the angles of a rhombus becomes equal, its each angle is a right angle.

It implies that it will have all angles equal to right angle and all sides are equal as well.

Thus, an equiangular rhombus is a square.

(b) A rectangle has all its angle equal to right angle.

Opposite sides are equal and parallel.

When all the sides become equal, it will be a square.

(c) A parallelogram has its opposite sides as  equal and parallel.

When all the angles become right angles, it can be a rectangle.

Hence, an equiangular rhombus and an equilateral rectangle can be a square but a parallelogram with four right angles cnnot be so.

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An angle bisector is a ray, segment, or line that divides a given angle into two angles of equal measure. Therefore, the correct answer is option C.

An angle is formed when two straight lines or rays meet at a common endpoint. The common point of contact is called the vertex of an angle.

An angle bisector is defined as a ray, segment, or line that divides a given angle into two angles of equal measures.

Steps to Construct an Angle Bisector:

Step 1: Draw any angle, say ∠ABC.

Step 2: Taking B as the center and any appropriate radius, draw an arc to intersect the rays BA and BC at, say, E and D respectively. (Refer to the figure below)

Step 3: Now, taking D and E as centers and with the same radius as taken in the previous step, draw two arcs to intersect each other at F.

Step 4: Join B to F and extend it as a ray. This ray BF is the required angle bisector of angle ABC.

Step 2: Taking B as the center and any appropriate radius, draw an arc to intersect the rays BA and BC at, say, E and D respectively. (Refer to the figure below)

Therefore, option C is the correct answer.

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Answer:

the answer is d

Step-by-step explanation:

everybody else keeps getting it wrong, the answer is d

Answer:  A flowchart proof presents a logical argument using boxes and arrows.

Step-by-step explanation:

We know that a flowchart proof is a way of representation of a mathematical proof that includes a logical series of statements ( or can say arguments) in boxes with connecting arrows.

i.e. we use boxes and arrows in a flow chart proof.

By the definition of flowchart proof, the complete sentence will be : A flowchart proof presents a logical argument using boxes and arrows.

The reasonable conjecture for Kathryn to make by recognizing a pattern and using inductive reasoning is:

  When a pair of lines intersect, the vertical angles are congruent.

The vertical angles are the opposite angles that are formed by the intersection of two lines. They are also known as Vertically opposite angles.

We know that when a pair of lines intersect then the vertical angles are always congruent.

However the measure of the angles may be acute or obtuse or right angle.

The same could be seen with the help of the figure attached to the answer.

Final answer:

The coefficient of the fourth term in the expansion of (x + 3)⁴ is calculated using the binomial theorem, which results in a coefficient of 108.

Explanation:

To find the coefficient of the fourth term in the expansion of (x + 3)⁴, we can use the binomial theorem. The binomial theorem states that when a binomial is raised to a power, the expansion consists of terms in the form of [tex]C(n, k) \times a^{n-k} \times b^k[/tex], where C(n, k) is the binomial coefficient. In our case a = x and b = 3, and n = 4.

For the fourth term where k = 3, the term would be [tex]C(4, 3) \times x^{4-3} \times 3^3[/tex]. The binomial coefficient C(4, 3) is equal to 4 because there are four ways to choose three items out of four. Therefore, the coefficient is 4 × 3³ = 4 × 27 = 108. So, the coefficient of the fourth term in the expansion of (x + 3)⁴ is 108.

The value of x in 5(x + 1) = 4(x + 8) is 27.

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides. LHS = RHS is a common mathematical formula.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given:

5(x + 1) = 4(x + 8)

Now, applying distributive property

5x+ 5 = 4x+ 32

5x - 4x = 32- 5

x= 27

Hence, the value of x is 27.

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Answer:

B. Angle PTR and angle PTS are supplementary angles.

Step-by-step explanation:

As R, T, S are collinear and PQ intersects RS at T, so ∠RTP and ∠STP are supplementary angles.

[tex]\Rightarrow m\angle RTP+m\angle PTS=180^{\circ}[/tex]  -------1

As P, T, Q are collinear and RQ intersects PQ at T, so ∠PTS and ∠QTS are supplementary angles.

[tex]\Rightarrow m\angle PTS+m\angle QTS=180^{\circ}[/tex]  ------2

Subtracting equation 1 and 2,

[tex]\Rightarrow m\angle RTP+m\angle PTS-m\angle PTS-m\angle QTS=180^{\circ}-180^{\circ}[/tex]

[tex]\Rightarrow m\angle RTP-m\angle QTS=0[/tex]

[tex]\Rightarrow m\angle RTP=m\angle QTS[/tex]

[tex]\Rightarrow m\angle PTR=m\angle STQ[/tex]

Therefore, to prove that angle PTR is always equal to angle STQ the statement needed is angle PTR and angle PTS are supplementary angles.

Final answer:

To divide 6.4 by 43.52, the approximate quotient is 0.09238.

Explanation:

To divide 6.4 by 43.52, you can simply perform the division operation. Here are the steps:

Therefore, 6.4 divided by 43.52 is approximately 0.09238.

Answer:

slope of perpendicular line = [tex]-\frac{1}{2}[/tex]

Step-by-step explanation:

Find  the slope of a line that is perpendicular to the line shown on the graph

To find slope of the given line , we pick two points from the graph

(0,1) and (1,3)

[tex]slope = \frac{y_2-y_1}{x_2-x_1} =\frac{3-1}{1-0} = 2[/tex]

Slope of given line = 2

slope of perpendicular line = negative reciprocal of slope of given line

slope of perpendicular line = [tex]-\frac{1}{2}[/tex]

1. where is the y intercept of any direct variation equation?

A direct variation equation is one in which the general form is:

y = k x

where k is the constant of proportionality

The y-intercept in this case is therefore zero so it is located on the origin (0, 0)

2. The equation which is a direct variation is:

                3) 3x-7y=0

                Because we can write this as:

                y = (3/7) x

3. what is the direct variation equation if y=8 and x = 4?

We solve for the value of k since we know that:

k = y/x

k = 8/4 = 2

So the equation is:

y = 2x

4. what is the direct variation equation if y =-5 and x = 7?
We also solve for the value of k since we know that:

k = y/x

k = -5/7

So the equation is:

y = (-5/7) x

Answer:                                  

The answer is trade association.                

Step-by-step explanation:              

Such organizations are founded to encourage trade and collaboration between companies. These are also known as industry trade group or business association and is funded by businesses that operate in a particular industry.          

If two chords are the same distance from the center of a circle then they are congruent.

Given that, two chords are the same distance from the center of a circle.

The chord of a circle is a line that connects two points that are located on the circumference of the circle. These two points can be located anywhere on the same circumference.

Chords are equidistant from the center if and only if their lengths are equal. Equal chords are subtended by equal angles from the center of the circle.

Hence, if two chords are the same distance from the center of a circle then they are congruent.

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[tex]\text{Answer: a) }P(\text{ getting an orange )}=\frac{7}{20}[/tex]

b) Blue color had an experimental probability that matched its theoretical probability.

Explanation:

Since we have given that

Number of times this spinner is spinned = 60

Number of times black occur = 17

Number of times blue occur = 15

Number of times orange occur = 21

Number of times purple occur = 7

a) So, Experimental probability of a spin of orange is given by

[tex]P(\text{ getting an orange }=\frac{\text{ Number of times orange occur}}{\text{total number of times the spinner spins}}\\\\P(\text{ getting an orange})}=\frac{21}{60}=\frac{7}{20}[/tex]

b) which color had an experimental probability that matched its theoretical probability.

According to theoretical probability ,

Every event must have equal probability, i.e. [tex]\frac{1}{4}[/tex]

And,

[tex]P(\text{ getting blue color)}=\frac{15}{60}=\frac{1}{4}[/tex]

So, Blue color had an experimental probability that matched its theoretical probability.

Final answer:

The expression 3√54−4√24 simplifies to √6 after factoring out the perfect squares under the radicals and combining like terms.

Explanation:

The student's question requires simplifying the expression 3√54−4√24. To simplify a square root, we look for perfect squares that are factors of the number under the root. For √54, we can notice that 54 = 9 * 6, and since √9 is a perfect square, we have 3√54 = 3*√9*√6 = 3*3*√6 = 9√6. Similarly, for √24, it is 24 = 4 * 6, so √24 = √4*√6 = 2*√6, and thus -4√24 = -4*2*√6 = -8√6. Combining these two results, we have 9√6− 8√6, which simplifies to √6. Therefore, the simplified expression is √6.