Find the distance between these points.
A (5,8), B(-3, 4)
AB =

Answer:

y=2x-14

Step-by-step explanation:

y=2x+8

y-y1=m(x-x1)

y-(-6)=2(x-4)

y+6=2x-8

y=2x-8-6

y=2x-14

Answer:

The value of A =  [tex]\dfrac{1}{16}[/tex]

The value of B = [tex]\dfrac{1}{16}[/tex]

The value of C =  [tex]\dfrac{25}{16}[/tex]

The value of D =  [tex]\dfrac{5}{4}[/tex]

Step-by-step explanation:

Given as :

The following relations are as

[tex]\dfrac{d}{5}[/tex] =4 A           ...........1

d = 20 B                                       ...........2

d = [tex]\dfrac{4}{5}[/tex] C           ...........3

And d = [tex]\dfrac{5}{4}[/tex]

Now, From the above relations

Put the value of d in eq 1

∵ 4 A = [tex]\dfrac{d}{5}[/tex]

So, d = 5 × 4 A

Or, [tex]\dfrac{5}{4}[/tex] = 20 A

∴ A = [tex]\frac{\frac{5}{4}}{20}[/tex] = [tex]\dfrac{1}{16}[/tex]

I.e The value of A =  [tex]\dfrac{1}{16}[/tex]

Again

From eq 2

∵ d = 20 B

Put the value of d

So, [tex]\dfrac{5}{4}[/tex]  = 20 B

Or, B = [tex]\frac{\frac{5}{4}}{20}[/tex]

∴ B = [tex]\dfrac{1}{16}[/tex]

I.e The value of B = [tex]\dfrac{1}{16}[/tex]

Similarly

From eq 3

d = [tex]\dfrac{4}{5}[/tex] C  

Put the value of d

So, [tex]\dfrac{5}{4}[/tex] =  [tex]\dfrac{4}{5}[/tex] C    

Or, C = [tex]\frac{5\times 5}{4\times 4}[/tex]

∴ C = [tex]\dfrac{25}{16}[/tex]

I.e The value of C =  [tex]\dfrac{25}{16}[/tex]

Hence The value are

The value of A =  [tex]\dfrac{1}{16}[/tex]

The value of B = [tex]\dfrac{1}{16}[/tex]

The value of C =  [tex]\dfrac{25}{16}[/tex]

The value of D =  [tex]\dfrac{5}{4}[/tex]

Answer

Answer:

The expression for Area of rectangle is [tex]x^2+16x+63[/tex].

Step-by-step explanation:

Given:

A rectangle has a length that is 4 units longer than the width.

Let the width of the rectangle be 'x' Units.

Length of rectangle will be = [tex]x+4\ Units[/tex]

If the width is increased by 7 units.

New Width of the rectangle = [tex]x+7\ Units[/tex]

Also the length increased by 5 units,

New Length of rectangle will be = [tex]x+4+5 = x+9\ Units[/tex]

We need to write an equivalent expression for area of the rectangle.

Area of rectangle is given by length times width.

framing in expression form we get;

Area of rectangle = [tex](x+7)\times(x+9)[/tex]

On Solving the equation we get;

Area of rectangle = [tex]x^2+9x+7x+63=x^2+16x+63[/tex]

Hence the expression for Area of rectangle is [tex]x^2+16x+63[/tex].

Answer:

[tex]y+6=\frac{5}{2}(x+2)[/tex]

y plus 6 equals StartFraction 5 Over 2 EndFraction left-parenthesis x plus 2 right parenthesis.

Step-by-step explanation:

we have the ordered pairs

[tex](-4,-11),(-2,-6),(6,14),(10,24)[/tex]

step 1

Find the slope

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

[tex]m=\frac{y2-y1}{x2-x1}[/tex]

take the points

[tex](-4,-11),(-2,-6)[/tex]

substitute the given values

[tex]m=\frac{-6+11}{-2+4}[/tex]

[tex]m=\frac{5}{2}[/tex]

step 2

Find the equation in point slope form

[tex]y-y1=m(x-x1)[/tex]

we have

[tex]m=\frac{5}{2}[/tex]

[tex](x1,y1)=(-2,-6)[/tex]

substitute in the equation

[tex]y-(-6)=\frac{5}{2}(x-(-2))[/tex]

[tex]y+6=\frac{5}{2}(x+2)[/tex]

Answer:

0 ≤ t ≤ 5.

Step-by-step explanation:

In the function [tex]f(t)[/tex], [tex]t[/tex] is the independent variable. The domain of [tex]f[/tex] is the set of all values of [tex]t[/tex] that this function can accept.

In this case, [tex]f(t)[/tex] is defined in a real-life context. Hence, consider the real-life constraints on the two variables. Both time and volume should be non-negative. In other words,

The first condition is an inequality about [tex]t[/tex], which is indeed the independent variable.

However, the second condition is about [tex]f[/tex], the dependent variable of this function. It has to be rewritten as a condition about [tex]t[/tex].

[tex]\begin{aligned} f(t) &\ge 0 &&\text{Assumption} \cr 80000 - 16000\, t& \ge 0 && \text{Definition of} ~ f \cr 80000 & \ge 16000\, t && \begin{aligned}&\text{Add $16000\, t$} \\[-0.5em] & \text{to both sides of the inequality}\end{aligned} \cr 5 &\ge t &&\begin{aligned}&\text{Divide both sides of} \\[-0.5em] & \text{the inequality by $16000$}\end{aligned} \cr t &\le 5 && \text{Flip the inequality}\end{aligned}[/tex].

Hence, t ≤ 5.

Combine the two inequalities to obtain the domain:

0 ≤ t ≤ 5.

Answer:

D. [tex]10\times 6^2[/tex]

Step-by-step explanation:

Please consider the complete question.

Nick has to build a brick wall. Each row of the wall requires [tex]6^2[/tex] bricks. There are 10 rows in the wall. How many bricks will Nick require to build the wall?

To find the number of total bricks required to make the wall, we will divide bricks used to make one row [tex](6^2)[/tex] by total number of rows that is 10.

[tex]\text{Number of total bricks required}=10\times 6^2[/tex]

Therefore, Nick will require [tex]10\times 6^2[/tex] bricks to make the wall and option D is the correct choice.

Answer:

angle bisector

Step-by-step explanation:

Definitions:

median: line that meets the middle of VX

angle bisector: line that cuts angle Y in half

altitude: line that meets VX at 90°

perpendicular bisector: line that meets VX at 90° and the middle of VX

YW does not meet VX at 90° and does not meet it in the middle. Without measurements or symbols, angle Y seems to be cut in half.

Answer:

Step-by-step explanation:

You can put the equation into intercept form by dividing by 30:

  -5x/30 + y/30 = 1

  x/(-6) + y/30 = 1

Intercept form is ...

  x/(x-intercept) +y/(y-intercept) = 1

This lets us recognize the x-intercept as -6 and the y-intercept as 30.

Work is provided in the image attached.

Final answer:

Using the ratio of 7 girls to 3 boys and knowing that there are 9 boys, we find that there are 21 girls at the movie by setting up a proportion and solving for the number of girls.

Explanation:

To find out the number of girls at the movie, we need to use the ratio given, which is 7 girls to 3 boys or 7:3. Since there are 9 boys, we can set up a proportion to solve for the number of girls (G):

7 girls : 3 boys = G girls : 9 boys

Now we cross multiply and solve for G:

7 * 9 = 3 * G

63 = 3 * G

Divide both sides by 3 to find the number of girls:

G = 63 / 3

G = 21

Therefore, there are 21 girls at the movie.

Answer:

25

Step-by-step explanation:

Do multiplication first 6*3 is 18 18+4 is 22 22+3 is 25

Answer:  Diameter is increasing as [tex]\frac{1}{3}[/tex] centimeter per hour.

Step-by-step explanation:

Alright, lets get started.

The formula for volume of sphere is given as V : [tex]\frac{4}{3}\pir^3[/tex]

The volume of a sphere is increasing at a rate of [tex]6\pi[/tex] cubic centimeters per hour.

It means : [tex]\frac{dV}{dt}=6\pi[/tex]

[tex]V=\frac{4}{3}\pi r^3[/tex]

Taking derivative with respect to t

[tex]\frac{dV}{dt}=\frac{4}{3}\pi \times 3r^2 \frac{dr}{dt}[/tex]

[tex]6\pi=4\pi r^2 \frac{dr}{dt}[/tex]

at the instant the radius of the sphere is 3 centimeters, means

[tex]\frac{dr}{dt}= \frac{6}{4 \times 3^2}[/tex]

[tex]\frac{dr}{dt}=\frac{1}{6}[/tex]

As [tex]radius = \frac{diameter}{2}[/tex]

[tex]\frac{1}{2}\frac{dD}{dt}=\frac{1}{6}[/tex]

[tex]\frac{dD}{dt} =\frac{1}{3}[/tex]

Hence diameter is increasing as [tex]\frac{1}{3}[/tex] centimeter per hour.   : Answer

Hope it will help :)

The volume of a sphere is increasing at a rate of 6π cubic centimeters per hour. The rate at which its diameter is increasing with respect to time at the instant with the radius of the sphere 3 centimeters is: [tex]\mathbf{\dfrac{1}{3}}[/tex]

Option A is correct.

The volume of a sphere can be represented by using the formula:

[tex]\mathbf{V = \dfrac{4}{3}\pi r^3}[/tex]

Now, by differentiation, if we differentiate the rate at which the volume is increasing with time, we have:

[tex]\mathbf{\dfrac{dV}{dt} = \dfrac{4}{3}\pi r^3 \ \dfrac{dr}{dt}}[/tex]

[tex]\mathbf{\dfrac{dV}{dt} = 4 \pi r^2 \ \dfrac{dr}{dt}}[/tex]

Given that:

Replacing the values into the differentiated equation, we have:

[tex]\mathbf{6 \pi= 4 \pi (3)^2 \ \dfrac{dr}{dt}}[/tex]

[tex]\mathbf{\ \dfrac{dr}{dt}=\dfrac{6 \pi}{4 \pi (3)^2}}[/tex]

[tex]\mathbf{\ \dfrac{dr}{dt}=\dfrac{6 }{4 \times 9}}[/tex]

[tex]\mathbf{\ \dfrac{dr}{dt}=\dfrac{1 }{6}\ cm/sec}[/tex]

∴

[tex]\mathbf{\dfrac{1}{2} \dfrac{dr}{dt} =\dfrac{1}{6} }[/tex]

[tex]\mathbf{ \dfrac{dr}{dt} =\dfrac{1}{6} \times \dfrac{2}{1}}[/tex]

[tex]\mathbf{ \dfrac{dr}{dt} =\dfrac{1}{3}}[/tex]

Therefore, we can conclude that the rate at which its diameter is increasing with respect to time at the instant with the radius of the sphere 3 centimeters is  [tex]\mathbf{\dfrac{1}{3}}[/tex]

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[tex]\bf (\stackrel{x_1}{-2}~,~\stackrel{y_1}{-6})\qquad (\stackrel{x_2}{2}~,~\stackrel{y_2}{2}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{2}-\stackrel{y1}{(-6)}}}{\underset{run} {\underset{x_2}{2}-\underset{x_1}{(-2)}}}\implies \cfrac{2+6}{2+2}\implies \cfrac{8}{4}\implies 2[/tex]

The slope of the line passing through the points (−2,−6) and (2,2) is 2, calculated using the slope formula in mathematics. This means there is a rise of 2 on the y-axis for every 1 increase on the x-axis.

Your task is to calculate the slope of a line passing through the points (−2,−6) and (2,2). In mathematics, slope is defined as the 'rise over run' - meaning, the change in the vertical axis (y-axis) for every unit increase in the horizontal axis (x-axis).

Conventionally, the slope (m) is calculated using the formula: m = (y2 - y1) / (x2 - x1). Applying this formula to the given points:

m = (2 - (-6))/(2 - (-2))

m = (2 + 6)/(2 + 2) = 8/4 = 2.

So, the slope of the line that passes through points (-2,-6) and (2,2) is 2; this means there's a rise of 2 on the y-axis for every 1 increase on the x-axis when you move from point (-2, -6) to point (2, 2).

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If [tex]P(4,7)\in G_f[/tex] where [tex]G_f=\{(x,f(x):\forall x\in\mathbb{R}\wedge f(x)\in\mathbb{R}\}[/tex] then [tex]f(x-2)[/tex] will shift the original function by 2 in the right resulting with new point [tex]P'(6,9)[/tex].

Hope this helps.

Answer:

AB ║ CD. (Proved)

Step-by-step explanation:

See the attached diagram of the triangle.

It is given that Δ ACD is an isosceles triangle.

Therefore, AC = AD and ∠ ACD = ∠ ADC, ⇒ ∠ 3 = ∠ 4 .......... (1)

Again, given that ∠ 1 = ∠ 3 ........... (2)

Now, from equations (1) and (2) we can write, ∠ 1 = ∠ 4

Now, AB and CD are two straight lines and AD is the transverse line and hence, ∠ 1 and ∠ 4 are alternate angles that are equal.

Therefore, AB and CD are parallel straight lines and AB ║ CD. (Proved)

Answer:

The probability that exactly 2 trucks are going over the speed limit is 56.3% (rounding to the next tenth)

Step-by-step explanation:

1. Let's check all the information provided to us to answer the question correctly:

Probability that a truck will be going over the speed limit on I-80 between Cheyenne and Rock Springs = 75%.

Number of trucks in the random sample = 5

2. What is the probability that exactly 2 trucks are going over the speed limit?

Let's recall that two events are independent if the result of the second event is not affected by the result of the 1st event. If a first truck goes over the speed limit and a second truck does the same are independent events, the probability of both events occurring is the product of the probabilities of the individual events.

P (T₁ and T₂) = P (T₁) * P (T₂)

P (T₁ and T₂ ) = 0.75 * 0.75 = 0.5625

The probability that exactly 2 trucks are going over the speed limit is 56.3% (rounding to the next tenth)

Answer:

b is the correct answer

The placebo effect describes how a person might show improvement simply because they believe a treatment is working, even if it's a dummy treatment. It's an essential aspect considered during clinical studies to distinguish between actual treatment effects and the impact of patients' expectations.

The placebo effect is best described as the phenomenon in which a person experiences a perceived improvement in their condition or symptoms simply due to their belief in the efficacy of the treatment, even if the treatment is inactive or a 'dummy' treatment, such as a sugar pill. This is represented in the provided choices by option B. Subjects may respond to a treatment simply because they believe it is affecting them.

Notably, this effect is considered in the design of clinical trials and studies. This is typically done by using a control group that receives a placebo treatment, which allows researchers to separate the physiological effects of the treatment from psychological effects induced by expectations- a crucial aspect of medical and scientific research.

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Answer:

Part A) Fisherman's Service will provide a better deal for a three days trip as it will cost $480 compare to the Wild Fishing Inc.'s deal that costs around 550$

Part B) Wild Fishing's Inc. would charge $700 for 6 days for both of them.

Part C) Let w be expression for Wild Fishing Inc. to represent the cost of the trip for both Mr. Martin and his son for n number of days trip.

w = 2(200) + 2n(25) = 400 + 50n

Let f be expression for Fisherman's Service to represent the cost of the trip for both Mr. Martin and his son for n number of days trip.

f = 2(150) + 2n(30) = 300 + 60n

Step-by-step explanation:

Part (A) Which company offers a better deal for a three day trip for Mr. Martin and his son? What will be the cost?

        Round Trip Flight $200 / Daily Rate $25

        Round Trip Flight $150 / Daily Rate $30

Analyzing Wild Fishing Inc. Offer for a three day trip

Expenses cost for Mr. Martin and his son if they chose Wild Fishing Inc

The formula to calculate Mr Martin expenses for 3 day trip:

Round trip Cost +  3 (Daily rate) ⇒ 200 + 3 (25) = $275.......[A]

The formula to calculate Mr Martin's son expenses for 3 day trip:

Round trip Cost +  3 (Daily rate) ⇒ 200 + 3 (25) = $275.......[B]

Adding Equation [A] and [B] will bring the net expense faced by Mr. Martin and his son if they chose to get Wild Fishing Inc. deal:

Net expense faced by Mr. Martin and his son: 275 + 275 = 550$

So, Wild Fishing Inc. will cost 550$ to both Mr. Martin and his son.

Analyzing Fisherman's Service Offer for a three day trip

As Mr. Martin and his son are planning a fishing trip to Alaska.

Expenses cost for Mr. Martin and his son if they chose Fisherman's Service

The formula to calculate Mr Martin expenses for 3 day trip:

Round trip Cost +  3 (Daily rate) ⇒ 150 + 3 (30) = $240.......[A]

The formula to calculate Mr Martin's son expenses for 3 day trip:

Round trip Cost +  3 (Daily rate) ⇒ 150 + 3 (30) = $240.......[B]

Adding Equation [A] and [B] will bring the net expense faced by Mr. Martin and his son if they chose to get Fisherman's Service deal:

Net expense faced by Mr. Martin and his son: 240 + 240 = 480$

So, Fisherman's Service will cost 480$ to both Mr. Martin and his son.

Choosing the right company that offers a better 3 days deal:

As we have determined that

Therefore, Fisherman's Service will provide a better deal for a three days trip as it will cost $480 compare to the Wild Fishing Inc.'s deal that costs around 550$.

Part (B) Which company would charge $700 for 6 days for both of them?

Checking Wild Fishing Inc. expenses for 6 days trip:

The formula to calculate Mr Martin expenses for 6 day trip:

Round trip Cost +  6 (Daily rate) ⇒ 200 + 6 (25) = $350.......[A]

The formula to calculate Mr Martin's son expenses for 3 day trip:

Round trip Cost +  6 (Daily rate) ⇒ 200 + 6 (25) = $350.......[B]

Adding Equation [A] and [B] will bring the net expense faced by Mr. Martin and his son if they chose to get Wild Fishing Inc. deal for 6 days:

Net expense faced by Mr. Martin and his son: $350 + $350 = 700$

Checking Fisherman's expenses for 6 days trip:

The formula to calculate Mr Martin expenses for 6 day trip:

Round trip Cost +  6 (Daily rate) ⇒ 150 + 6 (30) = $330.......[A]

The formula to calculate Mr Martin's son expenses for 3 day trip:

Round trip Cost +  6 (Daily rate) ⇒ 150 + 6 (30) = $330.......[B]

Adding Equation [A] and [B] will bring the net expense faced by Mr. Martin and his son if they chose to get Fisherman's Service deal for 6 days:

Net expense faced by Mr. Martin and his son: $330 + $330 = 680$

So, Wild Fishing's Inc. would charge $700 for 6 days for both of them.

Part (C) Write an expression to represent the cost of the trip for Mr. Martin and his son for each company.

Expression for Wild Inc.

Let w be expression for Wild Fishing Inc. to represent the cost of the trip for both Mr. Martin and his son for n number of days trip.

w = 2 (Round trip Cost) +  2n (Daily rate)

As Round trip costs $200 and $25 is the daily rate for Wild Fishing Inc.

So, w can be written as:

w = 2(200) + 2n(25) = 400 + 50n

Expression for Wild Inc.

Let f be expression for Fisherman's Service to represent the cost of the trip for both Mr. Martin and his son for n number of days trip.

f = 2 (Round trip Cost) +  2n (Daily rate)

As Round trip costs $150 and $30 is the daily rate for Fisherma''s Service.

So, w can be written as:

f = 2(150) + 2n(30) = 300 + 60n

Keywords: cost, expenses, deal

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Answer:

So it will take roughly 18.4471905815477 years.

(round appropriately if needed)

Step-by-step explanation:

A=P(1+r/n)^(nt)

6000=2000(1+0.06/4)^(4*t)

6000/2000 = (1+0.06/4)^(4*t)

3 = (1+0.06/4)^(4*t)

3 = (1.015)^(4*t)

log(3) = log((1.015)^(4*t))

log(3) = 4*t*log(1.015)

log(3)/(4*log(1.015)) = t

t = log(3)/(4*log(1.015))

t = 18.4471905815477

Answer:

C. f(x)=-3(x+7)(x+4)

Step-by-step explanation:

x-intercept means the value of x where the functions touches the x axis.

The function touches x axis at two places, x=-7 and x=-4 ie. both -7 and -4 are the roots of the equation.

And the terms (x+7) and (x+4) are the factors of the function f(x),

Only one such case exists where (x+7) and (x+4) are there as factors, ie. Option C.

Therefore, C.-3(x+7)(x+4).

Answer:

C. f(x)=-3(x+7)(x+4)

Step-by-step explanation:

Got it right on the test

Answer:

C

Lined up with the points

Answer:

The better bargain is the case of twelve 12 ounce cans.

Step-by-step explanation:

2-liter bottle of soda:

67.6 ounces cost $1.89, then

1 ounce costs

[tex]\$1.89\div 67.6\approx \$0.028[/tex]

A case of twelve 12 ounce cans:

[tex]12\times 12=144[/tex] ounces of the same soda cost $2.99, then

1 ounce costs

[tex]\$2.99\div 144\approx \$0.021[/tex]

The better bargain is the second choice, the case of twelve 12 ounce cans.

Answer:

2x^2-2x+2

Step-by-step explanation:

Solved through synthetic division

If you struggle with synthetic division you can use Math away a free calculator you can use on google.

After completing the synthetic division, the quotient in polynomial form is equal to: B. 2x² - 2x + 2.

Synthetic division can be defined as a simplified and specialized method that is used for the division of a polynomial by another polynomial of the first degree (x - n).

Also, you should use only the coefficients of the divided polynonial and its divisor and change the sign of the constant term in the divisor, in order to replace subtraction with addition.

Next, we would set up the synthetic division as follows:

-3 | 2       4         -4         6

   |_________________

      ↓     -6          6        -6

   |_________________

      2      -2          2         0

Quotient = 2    -2     2    ⇒  2x² - 2x + 2.

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The answer is 1 mile an hour

Answer:

  Matt is not correct

Step-by-step explanation:

The remainder from division by 9 will always be a number less than 9. The number 17 is not less than 9, so Matt made an error.

_____

Matt may have tried to find the remainder by adding the digits of the dividend:

  2 + 9 + 6 + 0 = 17

Matt needs to continue this process until he gets a single digit:

  1 + 7 = 8

8 is the remainder from the division.

Answer:

2

Step-by-step explanation:

There are three numbers on the first spinner.  Two are not even (1 and 3).

So there are 2 possible outcomes that work: 1 and 6 or 3 and 6.

Final answer:

There are two outcomes that meet the criteria of the question: not showing an even number on the first spinner and showing a 6 on the second spinner.

Explanation:

The question asks how many outcomes result in an odd number on the first spinner and show a 6 on the second spinner. The first spinner has three equal sectors labeled 1, 2, and 3, of which two are odd (1 and 3). The second spinner has four equal sectors labeled 3, 4, 5, and 6, but we are only interested in the outcome when the spinner shows a 6.

Since we want the first spinner to show an odd number, we ignore the 2, leaving us with 1 and 3 as the possible outcomes for the first spinner. For the second spinner, we are only interested in the outcome of 6. This means we multiply the two outcomes from the first spinner (1 and 3) by the one outcome from the second spinner (6), resulting in two possible outcomes: (1,6) and (3,6).

Therefore, the correct answer is there are two outcomes that do not show an even number on the first spinner and show a 6 on the second spinner.

Answer: Oliver's goal is to read 45 minutes more next month

Step-by-step explanation:

Firstly, we need to know how much in minutes is the [tex]10\%[/tex] of [tex]450 min[/tex]:

[tex]450 min(10\%)=(450 min)(\frac{10}{100})=45 min[/tex]

This means Oliver wants to read 45 minutes more next month. If we add this number to the 450 minutes he read this month, we have a total of 495 minutes.

Answer:

The goal is 945 minutes

Step-by-step explanation:

Answer:

$5903.39

Step-by-step explanation:

This can be solve using compound interest formula. The formula is:

[tex]F=P(1+r)^t[/tex]

Where

F is the future amount (what we are looking for)

P is the present amount (which is 4800)

r is the rate of compound interest per year, in decimal (3% per year, 3/100 = 0.03)

t is the time in years ( t = 7)

Now we substitute these values into the formula and find F:

[tex]F=P(1+r)^t\\F=4800(1+0.03)^7\\F=4800(1.03)^7\\F=5903.39[/tex]

So, Colin would have $5903.39 after 7 years, in his account.

Point is equidistant from points A and B because a point on a perpendicular bisector is equidistant from the endpoints of the segment it makes up.

The angle bisector theorem states that a triangle's opposite side is divided into two halves by an angle bisector that is proportional to the triangle's other two sides.

A point on a perpendicular bisector is equidistant from the endpoints of the segment it intersects, therefore if mACD = 90°, point D is equidistant from points A and B.

Any point on the perpendicular bisector is simply equal distance from both endpoints of the line segment on which it is drawn, according to the perpendicular bisector theorem.

The answer is that point  is equidistant from points A and B because a point on a perpendicular bisector is equidistant from the endpoints of the segment it makes up.

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To prove that point D is equidistant from the jungle gym and monkey bars (points A and B), we would utilise the geometrical concept of congruent triangles. This involves drawing line segments from D to A and D to B, and if those segments are congruent, then point D is equidistant from points A and B. The process would encompass drawing a perpendicular bisector from point D to segment AB.

To demonstrate that point D is equidistant from the jungle gym and the monkey bars, Beth would need to use specific concepts from geometry. First, she would need to identify points A and B as the jungle gym and monkey bars respectively. Point D is the location of the swings.  

Next, she would draw line segments from D to A and D to B. If line segment DA is congruent to line segment DB, this means they have the same length, hence point D is an equal distance from both points A and B (the jungle gym and the monkey bars).

This can be proved through congruent triangles if a perpendicular bisector is drawn from point D to line segment AB. Such concept echoes that a point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment. The resulting triangles would be congruent, meaning every part, including the sides DA and DB, would be the same.

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