The height of a triangle is 8 centimeters greater than the base. The area of the triangle is 356.5 square centimeters. Find the length of the base and the height of the triangle.

Answers in bold

====================================================

Explanation:

Always start the proof with what you are given. It seems silly to repeat the given information, but it's just how all proofs are formed. This is to build from those given statements to lead to what we want to prove on the last line.

The term "midpoint" refers to splitting a segment into two equal smaller pieces. So for instance, if Q is the midpoint of PR, then the pieces PQ and QR are equal. Each smaller piece is half as long as PR.

The transitive property is the idea if A = B and B = C, then A = C. Think of it like substitution. Or you can think of a set of dominoes: A knocks down B, which knocks down C. Therefore A knocks down C.

The addition property of equality is where we go from A = B to A+C = B+C. I've added C to both sides. Adding the same thing to both sides will not change the equation. It keeps things balanced. When going from line 4 to line 5, we added PQ to both sides.

In line 6, we've gone from PQ+PQ to 2PQ.

In line 7, we replace PQ with QR on the right hand side. We use the substitution property here (refer to line 3 where we found PQ = QR)

Line 8 uses the segment addition postulate. If A,B,C are collinear, then AB+BC = AC. The point B doesn't necessarily need to be the midpoint.

Line 9 uses the transitive property again (we combine lines 7 and 8 to lead to line 9).

The final line has us divide both sides by 2 to get what is shown in the table. This is the division property of equality.

------------------------

Shortcut using informal logic:

PQ = QR

QR is half as long as QS

Therefore, PQ is also half as long as QS

Answer:

The last reason for  [tex]PQ=\frac{1}{2}QS[/tex]  is division property of equality

Step-by-step explanation:

The reason for  [tex]PQ=\frac{1}{2}QS[/tex]  can be evaluated as follows    

For the statement [tex]Q[/tex] is the midpoint of [tex]\overline{PR}[/tex], the reason is given

For the statement [tex]R[/tex] is the midpoint of [tex]\overline{QS}[/tex], the reason is given

For the statement [tex]PQ=QR, QR=RS[/tex] , the reason is definition of midpoint

The midpoint of a line segment is the point that separates the line segment into two congruent parts.

For the statement [tex]PQ=RS[/tex] , the reason is transitive property

Transitive property says that if [tex]a=b[/tex], and [tex]b=c[/tex], then [tex]a=c[/tex]

In this statement [tex]PQ=QR, QR=RS[/tex], then [tex]PQ=RS[/tex]

For the statement [tex]PQ+PQ=PQ+RS[/tex] , the reason is addition property of equality

Addition property of equality says that if [tex]a=b[/tex], then [tex]a+c=b+c[/tex]

For the statement [tex]2PQ=PQ+RS[/tex] , the reason is combining of like terms of algebraic expression

Combining like terms is the process to add or to substract the coefficient of the terms

In this statement combining like terms is at the left side of the equation,  [tex]PQ+PQ=2PQ[/tex]

For the statement [tex]2PQ=QR+RS[/tex] , the reason is substitution property

Substitution property of equality says that if [tex]a=b[/tex], then [tex]a[/tex] replaces [tex]b[/tex] in any equation, in this statement [tex]QR[/tex] replaces [tex]PQ[/tex]

For the statement [tex]QR+RS=QS[/tex] , the reason is substitution property and definition of midpoint

In this statement [tex]QR+RS[/tex] replaces [tex]2PQ[/tex], and [tex]QS[/tex]  is the sum of two congruent parts  [tex]QR+RS[/tex]

For the statement [tex]2PQ=QS[/tex] , the reason is substitution property

In this statement [tex]2PQ[/tex] replaces [tex]QR+RS[/tex]

For the statement  [tex]PQ=\frac{1}{2}QS[/tex]  , the reason is division property of equality

Division property of equality says that if [tex]a=b[/tex], and [tex]c\neq 0[/tex], then [tex]\frac{a}{c} =\frac{b}{c}[/tex]

In this statement [tex]\frac{2PQ}{2} =\frac{QS}{2}[/tex] or can be reduced into [tex]PQ=\frac{1}{2}QS[/tex]

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By applying the exterior angle property, the measure of the exterior angle to ∠G in triangle EFG is equal to 120.7°

In Mathematics, the exterior angle property can be defined as a theorem which states that the measure of an exterior angle in a triangle is equal in magnitude to the sum of the measures of the two remote or opposite interior angles of that triangle.

In order to determine the value of ∠G in triangle EFG (ΔEFG), we would have to apply exterior angle property. Mathematically, the exterior angle property can be used to model triangle EFG (ΔEFG) and this is given by:

m<G = m<E + m<F

Substituting the given parameters into the formula, we have;

m<G = 84.3° + 36.4°.

m<G = 120.7°.

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From the question, we could give annotations for the number of tickets sold and the amount of money in the cashbox. We also find that there is 2 other elements related to the relationship between the tickets sold and the amount of money in the cashbox; the price of each ticket and the amount of money in the cashbox before any tickets are sold. Hence we could make some annotations:

m = money in the cashbox (after tickets are sold)

p = price of each tickets

x = number of tickets sold

c = money in the cashbox (before tickets are sold)

Based on the question, we know that the amount of money in the cashbox is dependeding on the number of tickets sold and the original amount of money in the cashbox before any tickets are sold. The relationship between the money in the cashbox and the number of tickets sold is depending on the price of each ticket. Based on this understanding, we could rewrite the relationship into such equation:

m = px + c ... (i)

Using data from the question, we could input the datas into equation (i):

m = px + c

320 = 20p + c ... (ii)

492.50 = 43 p + c ... (iii)

980 = 108 p + c ... (iv)

By using equation (ii) and (iv) we could find the value of p

320 = 20p + c

980 = 108p + c -

660 = 88p

p = 7.5 ... (v)

Using the finding value of p, we could find the value of c by using subtitution of equation (v) into equation (ii):

320 = 20p + c

320 = 20(7.5) + c

320 = 150 + c

c = 170 ... (vi)

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For the given conditions, the length of box=6.92 feet, width of box=3.46 feet and height of box=2.31 feet.

Each object in three dimensions takes up some space. The volume of this area is what is being measured. The space occupied within an object's boundaries in three dimensions is referred to as its volume. It is also referred to as the object's capacity.

Here,

area of material=72 feet²

V=lbh

l=2b

72=lb+2lh+2bh

substitute l as 2b,

72=2b²+4bh+2bh

36=b²+3bh

3bh=36-b²

h=(36-b²)/3b

v=lbh

v=2b*b*(36-b²)/3b

v=2b*(36-b²)/3

=(72b-2b³)/3

differentiate the v,

v'=1/3(72-6b²)

v'=0

72-6b²=0

6b²=72

b²=12

b=√12

b=3.46 feet

l=2b=6.92 feet

h=(36-b²)/3b

=(36-12)/3*3.46

h=2.31 feet

The box measures 6.92 feet long, 3.46 feet wide, and 2.31 feet tall under the specified circumstances.

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Answer:

[tex]W=2\sqrt{3}=3.46\; \sf ft\;(2\;d.p.)[/tex]

[tex]L=4\sqrt{3}=6.93\; \sf ft\;(2\;d.p.)[/tex]

[tex]H = \dfrac{4\sqrt{3}}{3}=2.31\; \sf ft\;(2\;d.p.)[/tex]

Step-by-step explanation:

If the length is twice the width:

Surface area of the box

[tex]\implies SA = WL + 2WH + 2LH[/tex]

[tex]\implies SA = W(2W) + 2WH + 2(2W)H[/tex]

[tex]\implies SA = 2W^2 + 2WH + 4WH[/tex]

[tex]\implies SA = 2W^2 + 6WH[/tex]

If the surface area is 72 ft²:

[tex]\implies 2W^2 + 6WH = 72[/tex]

[tex]\implies 6WH = 72 - 2W^2[/tex]

[tex]\implies H = \dfrac{72 - 2W^2}{6W}[/tex]

[tex]\implies H = \dfrac{36 - W^2}{3W}[/tex]

Volume of the box

[tex]\implies V = LWH[/tex]

[tex]\implies V = (2W)WH[/tex]

[tex]\implies V = 2W^2H[/tex]

Substitute the found expression for H into the formula for volume:

[tex]\implies V = 2W^2\left(\dfrac{36 - W^2}{3W}\right)[/tex]

[tex]\implies V = \dfrac{2W^2(36 - W^2)}{3W}[/tex]

[tex]\implies V = \dfrac{2W(36 - W^2)}{3}[/tex]

[tex]\implies V = \dfrac{72W - 2W^3}{3}[/tex]

[tex]\implies V =24W-\dfrac{2}{3}W^3}[/tex]

To find the value of W when the volume is at its maximum, first differentiate the equation for volume with respect to W:

[tex]\implies \dfrac{\text{d}V}{\text{d}W}=24-2W^2[/tex]

Now set the derivative to zero and solve for W:

[tex]\implies 24-2W^2=0[/tex]

[tex]\implies 2W^2=24[/tex]

[tex]\implies W^2=12[/tex]

[tex]\implies W=\sqrt{12}[/tex]

[tex]\implies W=2\sqrt{3}[/tex]

As length is twice the width:

[tex]\implies L=2 \cdot 2\sqrt{3}[/tex]

[tex]\implies L=4\sqrt{3}[/tex]

To find H, substitute the found value of W into the expression for H:

[tex]\implies H = \dfrac{36 - W^2}{3W}[/tex]

[tex]\implies H = \dfrac{36 - (2\sqrt{3})^2}{3(2\sqrt{3})}[/tex]

[tex]\implies H = \dfrac{36 - 12}{6\sqrt{3}}[/tex]

[tex]\implies H = \dfrac{24}{6\sqrt{3}}[/tex]

[tex]\implies H = \dfrac{4}{\sqrt{3}}[/tex]

[tex]\implies H = \dfrac{4\sqrt{3}}{3}[/tex]  

The y-intercept and x-intercept of the following linear equation is

x intercept

x intercept refers to the value of x when y is zero

−32x−3y=3

at y = 0

-32x - 3 * 0 = 3

-32x = 3

x = -3/32 = -0.09375

the coordinate is (-0.094, 0)

y intercept

y intercept refers to the value of y when x is zero

−32x −3y=3

at x = 0

0 - 3y = 3

y = -1

the coordinate is (0, -1)

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Answer:

The solution is 4 7/8 or 39/8

Step-by-step explanation:

[tex] \sf 3 \frac{5}{8} + 1 \frac{2}{8} \\ \\ = \sf (3 + 1) + ( \frac{5}{8} + \frac{2}{8} ) \\ \\ = \sf 4 + \frac{7}{8} \\ \\ = \sf 4 \frac{7}{8} [/tex]

Or

[tex]\sf 3 \frac{5}{8} + 1 \frac{2}{8} \\ \\ = \frac{29}{8} + \frac{10}{8} \\ \\ = \sf \frac{39}{8} [/tex]

Answer:

2:7 or [tex]\frac{2}{7}[/tex]

Step-by-step explanation:

The cone total is 4 and the total orders is 14

[tex]\frac{4}{14}[/tex] = [tex]\frac{2}{7}[/tex]

Answer:

0.559%

Step-by-step explanation:

The chance of winning equals the number of tickets bought divided by the total number of tickets

2 tickets are bought by the person, and 358 tickets are sold in total

2/358 would equal 0.00558659218..., and the percentage would be 0.558659...%
Rounding 0.558659 to the nearest thousandth's place would yield
0.559, so 0.559% would be the final answer

Part (d)

If xy > 0, then the two items x and y must have the same sign.

Examples where x,y are both positive

Examples where x,y are both negative

This places us in either the first quadrant or third quadrant (Q1 and Q3)

======================================================

Part (e)

If y < 0, then we have points like (2,-5) and (1,-7). The x coordinate doesn't matter as long as the y coordinate is negative.

Visually speaking we are below the x axis. This places the point in Q3, Q4, or on the negative y axis. The point cannot be on the x axis. You can think of this point as being underground or underwater.

======================================================

Part (f)

If x = 0, then the point is on the vertical y axis. Recall that y intercepts always occur when x = 0. Two examples would be (0,4) and (0,12).

Answer:

(d)  The set of all points in quadrants I and III (x and y have the same sign).

(e)  The set of all points below the x-axis.

(f)  The set of all points on the y-axis.

Step-by-step explanation:

The Cartesian plane is divided into four quadrants.  

Quadrant I is the upper right quadrant and the rest follow in a counterclockwise direction.

Part (d)

For xy > 0, x and y should either both be positive or both be negative, i.e. have the same sign.

Therefore, the description that satisfies the given condition is:

Part (e)

For y < 0, y is always negative.

Quadrants III and IV are below the x-axis.

Therefore, the description that satisfies the given condition is:

Part (f)

The only place where x = 0 is the y-axis.

Therefore, the description that satisfies the given condition is:

The equation of the line that passes through the point (-6, 2) and has slope of -1/2 is y = (-1/2)x - 1

The slope of the line = -1/2

The slope of the line is the change in y coordinates with respect to the change in x coordinates of the line

The slope intercept form

y = mx + b

Where m is the slope of the line

b is the y intercept form

The coordinates of the point = (-6, 2)

Substitute the values in the equation

2 = (-1/2)(-6) + b

2 = 3 +b

b = 2 -3

b = -1

The equation will be

y = (-1/2)x - 1

Hence, the equation of the line that passes through the point (-6, 2) and has slope of -1/2 is y = (-1/2)x - 1

The complete question is :

What is the equation of a line that passes through (-6, 2) and has a slope of -1/2 ?

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Answer:

third option

Step-by-step explanation:

| - 2x | = 4

the absolute value function always gives a positive value, however, the expression inside can be positive or negative , that is

- 2x = 4 ( divide both sides by - 2 )

x = - 2

or

- (- 2x) = 4

2x = 4 ( divide both sides by 2 )

x = 2

solutions are x = - 2 , x = 2

these are illustrated on the number line by solid circles at - 2 and 2

this is indicated on the third number line

Number line represents the solutions to |–2x| = 4 is third number line (x1 = -2, x2= 2).

Solving steps :

1. Calculate

Calculate the absolute value.

|-2x| = 4

Use |ab| = |a| * |b| to transform the expression.

|-2x|

|-2| * |x|

The absolute value of any number is always positive.

2 * |x|

We did it!

|-2x| = 4

2 * |x| = 4

2. Divide both sides

Divide both sides of the equation by 2.

2 * |x| = 4

2 * |x| ÷ 2 = 4 ÷ 2

Any expression divided by itself equals 1.

|x| = 4 ÷ 2

Calculate the quotient.

|x| = 2

3. Separate into possible cases

Use the absolute value definition to rewrite the absolute value equation as two separate equations.

|x| = 2

x = 2

x = -2

4. The equation has 2 solutions.

x1 = -2, x2= 2

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Answer:

75% - 20% = 55%

Step-by-step explanation:

In the given rectangular prism vector line IM and LM intersect each other. Option B is correct.

It is defined as a six-faced shape, a type of hexahedron in geometry. It is a three-dimensional shape. It is also called a cuboid.

Six faces, twelve sides, and eight vertices make up a rectangular prism (corners). Three of the 12 edges meet to create right angles at each vertex.

Since from all 12 edges vector line IM and LM meet to form right angles at each vertex.

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Thus, in the given rectangular prism vector line IM and LM intersect each other. Option B is correct.

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The trigonometric ratio (trig ratio) for the angle in this right-angled triangle should be expressed as fractions in simplest terms as follows:

Sin(M) = 21/121

Cos(L) = 10/11

Tan(K) = 21/100

In Mathematics, a trigonometric ratio (trig ratio) can be calculated by using this mnemonic SOHCAHTOA:

Sinθ = Opposite/Hypotenuse

Sin(M) = KM/ML

Sin(M) = √21/11

Taking the square of both the numerator and denominator, we have:

Sin(M) = (√21)²/11²

Sin(M) = 21/121

For the cosine of an angle in a right-angled triangle, we have:

Cosθ = Adjacent/Hypotenuse

Cos(L) = KL/ML

Cos(L) = 10/11

For the tangent of an angle in a right-angled triangle, we have:

Tanθ = Opposite/Adjacent

Tan(K) = KM/KL

Tan(K) = √21/10

Taking the square of both the numerator and denominator, we have:

Tan(K) = (√21)²/10²

Tan(K) = 21/100

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The probability that exactly 4 insects will survive is 0.8975

Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, 0 indicates impossibility of the event and 1 indicates certainty.

probability= sample space/total possible outcome

the total possible outcome for the insecticide to kill insect is 60%

If 4 will survive out of 13, it means 7 will be killed by the insecticide, the percentage is now 7/13×100 = 53.85%

Therefore the probability for 4 insect to survive is 53.85/60= 0.8975

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The height of the arrow at 2.5 seconds is 71 feet.

Given:

An arrow released from a bow, traveling towards a target 200 yards away,reaches a max height of 42.6 ft in only 1.5 seconds.

1.5 seconds = 42.6 feet

divide by 1.5 on both sides.

1 sec = 28.4 feet

0.5 sec = 28.4/2

0.5 sec = 14.2 feet

2.5 sec = 1.5 sec + 1 sec

= 42.6 + 28.4

= 71 feet

Therefore The height of the arrow at 2.5 seconds is 71 feet.

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Answer:7/10

Step-by-step explanation:

Answer:

seven multiplied by x to the second power multiplied by the difference of two multiplied by x subtract 3 multiplied by the product of three multiplied by z multiplied by 3 add four.

Step-by-step explanation:

This is a useless question, word form isn't in any sort of ACTUAL GLOBAL TEST. So don't stress over this.

Answer:Marsha earned interest = $850

Step-by-step explanation:

Amount deposited = $8500

Time = 2 years

Interest rate = 5% = 0.05

We need to find how much did Marsha earn in interest

The formula use is:

Where I is interest earned, P is amount deposited, r is interest rate and t is time.

Putting values and finding interest.

So, Marsha earned interest = $850

If the angle elevation to the plane A = 30 degrees and B = 40 degrees , then the Airplane is 359.11 feet high .

In the question ,

it is given that

the distance between the two observer = 1050 feet

the measure of angle A is = 30°

the measure of angle B is = 40°

By angle sum property angle C is = 180 - (30 + 40)

= 180 - (70)

= 110

By Sine Law .

Sin(110)/1050 = Sin(30)/a = Sin(40)/b

Using , Sin(110)/1050 = Sin(30)/a ,

we get ,

Sin(110)/1050 = 0.5/a

a = 558.69

Using Sin(110)/1050 = Sin(40)/b

b = 718.24

By using Sine Law

Sin(40)/h = Sin(90)/558.69

h = 359.11

Therefore , If A = 30 degrees and B = 40 degrees , then the Airplane is 359.11 feet high .

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If triangle CDE is dilated by a scale factor of 1/4 to form triangle C'D'E' , then measure of the side DE is 14.

In the question ,

it is given that ,

The triangle CDE is dilated by scale factor of 1/4

and the new triangle formed after dilation is C'D'E' .

it is given that the measure of the side D'E' = 3.5 .

let the measure of the the side DE = x

So , according to the question ,

x *(1/4) = 3.5

On cross multiplication,

we get ,

x = 4* 3.5

x = 14

Therefore , If triangle CDE is dilated by a scale factor of 1/4 to form triangle C'D'E' , then measure of the side DE is 14.

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The length of the rectangle is 56.9 unit and the breadth of the rectangle is 25.9 unit.

Area of the rectangle is 1473.7 sq. unit.

Given, a rectangle's length is 5.1 more than 2 times its width.

Its perimeter is 165.6.

Let the length of the rectangle be l,

breadth of the rectangle be b.

According to the question,

l = 5.1 + 2b

Using the formula of Perimeter,

Perimeter = 2(l + b)

165.6 = 2(5.1 + 2b + b)

165.6 = 2(5.1 + 3b)

82.8 = 5.1 + 3b

3b = 77.7

b = 25.9

Now, l = 5.1 + 2(25.9)

l = 5.1 + 51.8

l = 56.9

So, the length of the rectangle is 56.9 unit

and the breadth of the rectangle is 25.9 unit

Now, the area of the rectangle be,

Area = l×b

Area = 56.9×25.9

Area = 1,473.71

Area of the rectangle = 1473.7 sq. unit

Hence, the length of the rectangle is 56.9 unit and the breadth of the rectangle is 25.9 unit.

Area of the rectangle is 1473.7 sq. unit.

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Answer:

27.64

Step-by-step explanation:

The number of cards Camila sold is 50cards

A percentage is a number or ratio that can be expressed as a fraction of 100. If we have to calculate percent of a number, divide the number by the whole and multiply by 100. Hence, the percentage means, a part per hundred. The word per cent means per 100. It is represented by the symbol “%”. For example if 10% of 200 students in a school are boys, the the number of boys in the school is 10/100× 200= 20students

Similarly Camila sold 10% of the card sold on mother's day. The number of cards sold on mother's day is 500.

Therefore Camila sold 10/100×500= 50cards

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Answer:

Step-by-step explanation:

The possible dimensions of the rectangular garden with an area of 72 square yards are (4,18), (8,9), (6,12), (2,36)

The measurement that expresses the size of a region on a plane or curved surface is called area. Surface area refers to the area of an open surface or the boundary of a three-dimensional object, whereas the area of a plane region or plane area refers to the area of a form or planar lamina.

Area is the entire amount of space occupied by a flat (2-D) surface or an object's shape. On a sheet of paper, draw a square using a pencil. It has two dimensions. The area of a shape on paper is the area that it occupies. Consider your square as being composed of smaller unit squares.

Given,

The area of a rectangular garden is 72 square yards.

And also given that the dimensions of the garden are not prime numbers.

LCM of 72 is,

=2×2×2×3×3

Therefore, the possible dimensions of the garden are:

(4,18)

(8,9)

(6,12)

(2,36)

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Answer: tell me the soulation like how do you solve it then I can help

Step-by-step explanation:

The total annual premium for this insurance will be $14,219.

Define premium.

An amount that the insured pays on a regular basis to the insurer to cover his risk is known as a premium. In terms of finance, a premium is the sum of the costs associated with an option or the distinction between the higher price paid for a fixed-income instrument and the relevant face amount. For many different insurance policies, including health, homeowner's, and rental insurance, premiums are typically paid. Automobile insurance is a typical example of an insurance premium.

Solution explained:

A/Q, as we calculate,

Annual Premium for one worker is = [tex]33000 X \frac{10.51}{100}[/tex] = $3468.3

Annual Premium for four workers is = 4 X 3468.3 = $13873.2

Annual Premium for the secretary is = 26000 X [tex]\frac{1.33}{100}[/tex] = 345.8

Therefore, the total annual premium for this insurance will be

= 13873.2 + 345.8 = 14219

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Answer:

1,148

Step-by-step explanation:

The error is that the student used the formula for a reflection in the x-axis, and the correct endpoints are: C'(1, 1) and D'(3, -2)

Rotation simply means turning.

In a cartesian plane, if a point P, whose position vector is (x,y), is rotated through 270° about the origin, the position of the image P' is (y, -x)

But the reflection in the x-axis is defined as (x,y) => (x, -y) i.e. if a position vector of a point is (x,y), the position vector of its image in the x-axis is  (x, -y)

Given: C(-1, 1) and D'(2, 3)

Therefore, the student used the formula for a reflection in the x-axis. The 2nd option is the answer.

The correct endpoints are: C'(1, 1) and D'(3, -2)

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The distance formula used to determine if the figure is a parallelogram indicates;

A parallelogram is a quadrilateral that have two pairs of facing sides that are parallel and of the same length.

The length of the sides can be found by using the formula for finding the distance between two points on the coordinate plane as follows;

The distance between the points (x₁, y₁) and (x₂, y₂) is d = √(x₂ - x₁)² + (y₂ - y₁)²

The length of QR = √((1 - (-10))² + (-1 - (-2))²) = √(122) ≈ 11.0

Length of ST = √((1 - (-11))² + (-7 - (-8))²) = √(145) ≈ 12.0

Length of QS = √((1 - (-10))² + (-7 - (-2))²) = √(146) ≈ 12.1

Length of RT = √((1 - (-11))² + (-1 - (-8))²) = √(193) ≈ 13.9

The lengths of the opposite sides of a parallelogram are the same. The lengths of the sides of the quadrilateral QRST are different, therefore, quadrilateral QRST is not a parallelogram.

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