In one of his experiments conducted with animals, Thorndike found that cats learned to escape from a puzzle box is increased gradually
To quantify the learning process, Thorndike used a mathematical formula known as the Law of Effect equation. The equation is:
B = f(log S1/S2)
where B represents the strength of the behavior, S1 represents the satisfaction of the positive consequence, and S2 represents the degree of frustration or negative consequence.
In the context of Thorndike's puzzle box experiment, the Law of Effect equation can be used to describe how the cat's behavior changed over time as it learned to escape the puzzle box more quickly and efficiently. Initially, the cat's behavior was weak because it did not know which actions would lead to a positive outcome.
To know more about puzzle box here
https://brainly.com/question/30470834
#SPJ4
The table represents some points on the graph of a linear function.
Which equation represents the same relationship?
y= -1.5x+3
y = 3x - 1.5
y= 1.5x -3
y= -3x+1.5
An equation that represents the same relationship include the following: D. y = -3x + 1.5.
How to determine an equation of this line?In Mathematics, the point-slope form of a straight line can be calculated by using the following mathematical expression:
y - y₁ = m(x - x₁) or [tex]y - y_1 = \frac{(y_2- y_1)}{(x_2 - x_1)}(x - x_1)[/tex]
Where:
m represent the slope.x and y represent the points.At data point (-3.5, 10.5), a linear equation in slope-intercept form for this line can be calculated by using the point-slope form as follows:
[tex]y - y_1 = \frac{(y_2- y_1)}{(x_2 - x_1)}(x - x_1)\\\\y - 10.5 = \frac{(3- 10.5)}{(-0.5 +3)}(x +3)[/tex]
y - 10.5 = -3(x + 3)
y = -3x - 9 + 10.5.
y = -3x + 1.5
In this context, we can reasonably infer and logically deduce that a linear function of the line that represents this table in slope-intercept form is y = -3x + 1.5.
Read more on slope here: brainly.com/question/3493733
#SPJ1
use the methods taught in this class to find the slope and y-intercept of the regression line. round your answers to 3 decimal places.
The slope and Y-intercepts of the regression lines can be found using the formulae: m = (nΣxy - ΣxΣy) / (nΣx² - (Σx)²) and b = (Σy - mΣx) / n
To find the slope and y-intercept of a regression line, you can use the formula y = mx + b, where m is the slope and b is the y-intercept. Regression Line is a method in which the dependent variable serves as the explanatory variable, and the former is utilized to infer information about the latter and create a forecast for it.
To calculate the slope of a regression line, you can use the formula m = (nΣxy - ΣxΣy) / (nΣx² - (Σx)²), where n is the number of data points, x and y are the variables, and Σ represents summation.
To calculate the y-intercept of a regression line, you can use the formula b = (Σy - mΣx) / n.
Once you have calculated these values, you can substitute them into the equation y = mx + b to get your regression line.
To know more about slopes and Y-intercepts, refer:
https://brainly.com/question/9753782
#SPJ4
Correct question is:
Use the methods taught in this class to find the slope and y-intercept of the regression line. Round your answers to 3 decimal places.
The box plots show a random sample of wait times for two rides at a water park
The median wait time for Speed Slide is 2 minutes longer than the median wait time for Wave Machine and the IQR for both rides is 6 minutes.
Define the term box plot?A box plot, also known as a box and whisker plot, is a graphical representation of a data set that shows the distribution of the data along a number line.
In the box plots show a random sample of wait times for two rides at a water park is shown in figure.
If we compare the wait times in the box plots,
then for, Speed Slide: Median = 11
IQR= Q1 - Q3 (Calculation formula of IQR)
= 12 - 6
IQR = 6 minutes
for wave slide: median: 9
IQR= Q1 - Q3 (Calculation formula of IQR)
= 11 - 9
IQR = 2 minutes
The median wait time for Speed Slide is 2 minutes longer than the median wait time for Wave Machine and the IQR for both rides is 6 minutes.
To know more about median, visit:
https://brainly.com/question/14532771
#SPJ1
Helpp with these questions please
Solution In the Attachment Above
Hope It Helps :)
helps asap !!!!!
if the original price of a refrigerator is $3200 tiana buys the refrigerator at a sale for 20% off the original price she has a discount voucher that gives her a further $64 off the sale price
what percentage of the original price does tiana pay for the refrigerator ?
(can somebody show me how to do this step by step)
Answer:
she payed 78% of the origional price
Step-by-step explanation: ok so boom right 20% of 3200 is 640 so that would make 2% 64 which would be the discount voucher ( 20% = 640 take a away a 0 on both and it shows u) so u do the 20% plus the 2% and that would make 22% and 22-100 is 78%. :)
PLS HELP 40 POINTS AND BRAINLIEST!!
Step-by-step explanation:
All of the EXTERNAL angles of a polygon sum to 360 °
95 + 44 + 42 + 78 + p = 360°
p =101°
then q + p form a straight line = 180 °
so q = 79 °
write the equation for the relationship between x and y. x=1,2,4 y= 0,-5,-15
Answer:
y = -5x + 5
Step-by-step explanation:
y = mx + b
y = __x + __
To write the equation we need the slope (m) and the y-intercept (b).
Slope:
The is the change in y over the change in x. As the y is decreeing by 5, the x is increasing by 1. This gives us the slope -5/1 = -5
y-intercept:
The y intercept is the point (0,b). It is when x equals zero. If we start at 1 and go back to 0. That mean that the y increase by 5, so the intercept is at the point (0,5) which makes the y-intercept 5.
y = -5x + 5
Helping in the name of Jesus.
How do you compute the sum of squared errors
Answer:
Relating SSE to Other Statistical Data
Variance = SSE/n, if you are calculating the variance of a full population.Variance = SSE/(n-1), if you are calculating the variance of a sample set of data.
FILL IN THE BLANK ______ can creep into a study if subjects are picked (intentionally or not) according to some criterion and not randomly.
Selection bias can creep into a study if subjects are picked (intentionally or not) according to some criterion and not randomly.
Selection bias occurs when the selection of study participants is not random and is instead influenced by certain criteria. This can result in a non-representative sample that does not accurately reflect the population being studied, leading to inaccurate conclusions. For example, if a study on the effectiveness of a medication only enrolls participants who are already known to respond well to that medication, the results may overestimate its effectiveness in the general population. To minimize selection bias, researchers should use random sampling techniques and carefully consider the inclusion and exclusion criteria .
To know more about Selection bias, here
brainly.com/question/30355124
#SPJ4
0 / 1
It is November 1 of Year 1. Sales for Dee Company for November and December of Year 1 and January of Year 2 are forecasted to be as follows:
November, 400,000; December 600,000; January, 200,000
60% of sales are credit sales; the remaining sales are cash sales. Of these credit sales, 5% are collected during the month of sale, 25% in the following month, and 65% in the second following month; 5% are never collected. Total sales for September and October of Year 1 were 100,000 and 150,000, respectively.
What is the forecasted amount of total cash collections in November of Year 1?
Answer:
Step-by-step explanation:
To find the forecasted amount of total cash collections in November of Year 1, we first need to determine the credit sales for September, October, November, and December of Year 1, and January of Year 2.
Credit sales for September and October of Year 1 have already been given as 100,000 and 150,000, respectively.
For November, credit sales are 60% of 400,000 = 240,000.
For December, credit sales are 60% of 600,000 = 360,000.
For January of Year 2, credit sales are 60% of 200,000 = 120,000.
Now, we need to calculate the amount of credit sales that will be collected in November of Year 1.
5% of credit sales for November will be collected in November itself, which is 5% of 240,000 = 12,000.
25% of credit sales for November will be collected in December, which is 25% of 240,000 = 60,000.
65% of credit sales for November will be collected in January of Year 2, which is 65% of 240,000 = 156,000.
The remaining 5% of credit sales for November, which is 5% of 240,000 = 12,000, will never be collected.
Therefore, the total cash collections in November of Year 1 will be the sum of cash sales for November, and the amount of credit sales collected in November:
Cash sales for November = 40% of 400,000 = 160,000
Credit sales collected in November = 12,000
Total cash collections in November of Year 1 = 160,000 + 12,000 = 172,000.
Hence, the forecasted amount of total cash collections in November of Year 1 is $172,000.
Michelle and Robert constructed a wooden bridge for ATVs over a deep creek on the edge of their property. To recuperate the cost of the materials, they decided to charge an annual toll of $8 to each of the 120 members of a local club. A survey showed that for every $1 the toll is increased, 4 members wouldn't use the bridge anymore. What is the best toll charge to allow them to recuperate the cost of the materials the fastest?
Answer:
the answer is $10
Step-by-step explanation:
trust that
The cost white in dollars for X pounds of deli meat is represented by the equation Y equals 3.5 X graph the equation and interpret the slope
The graph is of the given equation is represented in the figure below.
Define the term graph?The visual representation of mathematical functions or data points on a Cartesian coordinate system is an x-y axis graphic.
By dividing the price change by the fluctuation in the amount of deli meat, one may calculate the slope. The magnitude indicates that the price per pound of deli meat has changed by 3.5 units.
Given linear equation represents the cost measured in dollars, as a function of the amount of deli meat, measured in pounds:
[tex]y = 3.5x[/tex]
The graph is of that linear equation as plotted below diagram.
By dividing the price change by the fluctuation in the amount of deli meat, one may calculate the slope. The magnitude denotes a change in the price per pound of deli meat of 3.5 units.
To know more about graph equation, visit:
https://brainly.com/question/29464771
#SPJ1
The graph is of the given equation is represented in the figure below linear line: Y = 3.5x
Define the term graph?The visual representation of mathematical functions or data points on a Cartesian coordinate system is an x-y axis graphic.
By dividing the price change by the fluctuation in the amount of deli meat, one may calculate the slope. The magnitude indicates that the price per pound of deli meat has changed by 3.5 units.
Given linear equation represents the cost measured in dollars, as a function of the amount of deli meat, measured in pounds:
Linear line: Y = 3.5x
The graph is of that linear equation as plotted below diagram.
By dividing the price change by the fluctuation in the amount of deli meat, one may calculate the slope. The magnitude denotes a change in the price per pound of deli meat of 3.5 units.
To know more about graph equation, visit:
brainly.com/question/29464771
#SPJ1
Fredericka Smith's account statement. Unpaid balance of $75.06. Periodic rate of 2 percent. What is the finance charge? New purchases of $432.11. What is the new balance?
Answer:
Step-by-step explanation:
$75.06×2%=$75.06×0.02=$1.50
The finance charge is the product of the unpaid balance and the periodic rate.
Graph the system of linear equations.
4x + 3y = 24
-2x + 6y = 18
Use the Line tool to graph the lines.
10. Audry spends $600 per month on rent. She makes $25,000 per year. To the nearest percent, what percent of her income does she spend on rent? F 28% G 21% H 20% J 29%
Answer:
J 29%
Step-by-step explanation:
Monthly rent: $600.
There are 12 months in 1 year, so the total rent she pays in 1 year is
12 × $600 = $7200
We need to find what percent $7200 is of $25,000.
percent = part/whole × 100%
percent = 7200/25000 × 100%
percent = 28.8%
Rounded to the nearest percent, the answer is 29%.
3.
h
b=32 inches
h = 22 inches
Find the area of the triangle above.
O242 sq. inches
O125 sq. inches
O 300 sq. inches
O 352 sq. inches
There are 352 square inches in the triangle's surface area. Hence, O 352 sq. inches is 's correct response.
The correct answer is 352 sq. inches
Which fundamental principle of a triangle is this?The triangle has the following characteristics: There are 180 angles in a triangle, thus they all sum up to 180. The lengths of the two sides of a triangle are greater than the length of the third side. Similar to this, the third side of a triangle has a shorter length than the angle between its 2 sides.
A = (1/2)bh if b is the quadrilateral base and h is its height.In this instance, the triangle's height is 22 inches, while its base is 32 inches. These values are substituted into the calculation to produce the following result: A = (1/2)(32)(22) A Equals 352 square inches
To know more about triangle visit:
https://brainly.com/question/2773823
#SPJ1
3x + y = 6
Y + 2 = x
Answer: x = 2, y = 0
Step-by-step explanation:
Assuming you need help solving for x or y, and the capital Y is y, we have the system of equations:
3x + y = 6
y + 2 = x
Substituting x for y + 2 gives us
3(y + 2) + y = 6
3y + 6 + y = 6
4y = 0
y = 0
Plugging y = 0 in for the second equation gives us
x = 0 + 2, or x = 2
5x+6/1/3=3x/0.4 HELP RSM.
Answer:
x = 0.8
Step-by-step explanation:
5x+6/1/3=3x/0.4
5x + 2 = 3x/0.4
2x + 0.8 = 3x
x = 0.8
Keekee bought a magazine for $6 and some binders for $9 each. She spent a total of $78. How
many binders did she buy? Step by step Please!!
Answer:
10 binders ........... .......... .....
Answer:
For this one I assigned y as the missing number
We know that $6 magazine plus $9 times a specific amount would equal $78
so 6 + 9(y) = 78
6 + 9(8) = 78
9(8)=72
72+6=78
So there were 8 binders bought at $9 each and a 1 magazine bought for $6.
Step-by-step explanation:
Classify each statement according to whether it describes a regression or a coefficient of determination Regression Coefficient of determination represented as R models the relationship between two variables describes how well data points fit a line values range from 0 to 1 can be positive or negative can be described by the linear equation YmX + b
The claims under "Regression" are as follows: 1. can be described by the l
Regression is a statistical technique that connects one or more independent (explanatory) variables to a dependent variable. If there is a relationship between changes in one or more of the explanatory factors and changes in the dependent variable, it can be shown using a regression model.
Y = mx + b is how the slope-intercept form of the equation for a straight line is expressed.
In the equation y = mx + b, m denotes the slope of the line and b its intercept. The distance between the line and the x- and y-axes is denoted by the letters x and y, respectively.
The first of the following claims, "Is used to assess the fit of a model," is true.
b) By including more factors, the result may be overstated.
called the coefficient of determination.
d.) Indicates the proportion of the variation in y that the model can account for.
e.) It is equivalent to the correlation coefficient r2 in basic linear regression.
What is Regression Analysis?
A collection of statistical procedures known as regression analysis is used to estimate the correlations between a dependent variable and one or more independent variables.
SS(res) + SS(reg) = SS, R2 1 - SS(res) / SS(tot) (tot)
R2 is defined as (SS(reg)/SS(tot) = (SS(reg)/n)/(SS(tot)/n)
Where SS(tot) = (y - y(bar)), the total sum of squares (proportional to the variance of the data), 2 • The explained sum of squares, also known as the regression sum of squares, is SS(reg) = (f(i)-y(bar)). SS(res) = (y(i) - f(i))2 = e2 is the sum of squares of residuals, often known as the residual sum of squares (i)
To know more about Regression, click the below link
https://brainly.com/question/29855763
#SPJ4
I need help I really don’t get this
The result of multiplying [tex]5,230[/tex] by [tex]25[/tex] is a quotient of [tex]209.2[/tex] rupees.
What in arithmetic is a quotient?In this example, the number being divided (15) is known as the reward, and the number being divided by (3 in this instance) is known as the divisor. The consequence of divide is the quotient.
What is the ratio between two numbers?When we divided one number by another, the result is a quotient. As an instance, when we divide 6 with 3, we obtain the answer 2, which corresponds to the division.This quotient may be either an integer or a decimal. For describing the presence or intensity of a quality in someone or something, the quotient is utilized.
25)5230(209.2
-50
--------------
230
-225
-----------------
50
-50
------------
0
-----------
So the quotient is [tex]209.2[/tex]
To know more about quotient visit:
https://brainly.com/question/16134410
#SPJ1
PLEASE HELP AND SHOW THE WORK
An equation of the line that goes through the point (-1, -3) and (3, 5) is y = 2x - 1.
An equation of the line in slope-intercept form that is perpendicular to the equation for obstacle 1 is y = -x/2 + 3.
How to determine an equation of this line?In Mathematics, the point-slope form of a straight line can be calculated by using the following mathematical expression:
y - y₁ = m(x - x₁) or [tex]y - y_1 = \frac{(y_2- y_1)}{(x_2 - x_1)}(x - x_1)[/tex]
Where:
m represent the slope.x and y represent the points.At data point (-1, -3), a linear equation in slope-intercept form for this line can be calculated by using the point-slope form as follows:
[tex]y - y_1 = \frac{(y_2- y_1)}{(x_2 - x_1)}(x - x_1)\\\\y - (-3) = \frac{(5- (-3))}{(3-(-1))}(x -(-1))\\\\y +3 = \frac{(5+3)}{(3+1)}(x +1)[/tex]
y + 3 = 2(x + 1)
y = 2x + 2 - 3
y = 2x - 1
In Mathematics, a condition that must be met for two lines to be perpendicular is given by:
m₁ × m₂ = -1
2 × m₂ = -1
m₂ = -1/2.
At point (-4, 5), an equation of the line can be calculated by using the point-slope form as follows:
y - y₁ = m(x - x₁)
y - 5 = -1/2(x + 4)
y = -x/2 - 2 + 5
y = -x/2 + 3
Read more on point-slope here: brainly.com/question/24907633
#SPJ1
Find x, if √x +2y^2 = 15 and √4x - 4y^2=6
pls help soon
Step 1: Rearrange the equation √x +2y^2 = 15 to get x = (15 - 2y^2)^2.
Step 2: Substitute x = (15 - 2y^2)^2 in the equation √4x - 4y^2=6.
Step 3: Simplify the equation to get 4(15 - 2y^2)^2 - 4y^2 = 6.
Step 4: Solve for y^2 by rearranging the equation to get y^2 = (6 + 4(15 - 2y^2)^2)/8.
Step 5: Substitute y^2 = (6 + 4(15 - 2y^2)^2)/8 in the equation x = (15 - 2y^2)^2.
Step 6: Solve for x by rearranging the equation to get x = (15 - (6 + 4(15 - 2y^2)^2)/4)^2.
Step 7: Substitute y^2 = (6 + 4(15 - 2y^2)^2)/8 in the equation x = (15 - (6 + 4(15 - 2y^2)^2)/4)^2.
Step 8: Simplify the equation to get x = (15 - (6 + 60 - 8y^
Which equation defines y as a linear function of x?
Select all that apply.
□ A. y = 3x² +7
□ B. x - 4y = 5
□ C. 5x² + y = 3
OD. 2(x+3y) = 6
O E. y = 5x - 3x + 4
Among the given equations, the equation E. y = 5x - 3x + 4 is the only equation that is in the form y = mx + b.
What is linear equation ?
A linear function is defined as a function that has a constant rate of change, meaning that as x increases by a certain amount, y also increases by a certain amount. Therefore, the equation that defines y as a linear function of x is of the form y = mx + b, where m is the slope and b is the y-intercept.
Among the given equations, the equation E. y = 5x - 3x + 4 is the only equation that is in the form y = mx + b. By combining like terms, we can simplify it to y = 2x + 4. This equation has a constant rate of change of 2, which means that for every 1 unit increase in x, y will increase by 2 units.
The other equations are not linear functions because they either have an x² term, which causes a non-constant rate of change, or they have multiple variables, which means that changes in x will not produce a consistent change in y.
Therefore, the answer is E. y = 5x - 3x + 4.
To know more about equations visit :-
https://brainly.com/question/22688504
#SPJ1
Part A
Use GeoGebra to graph points A, B, and C to the locations shown by the ordered pairs in the table. Then join each pair of
points using the segment tool. Record the length of each side and the measure of each angle for the resulting triangle.
Location
A(3,4), B(1,1).
C(5.1)
A(4.5), B(2.1).
C(7.3)
—————-
AB=
BC=
AC=
Answer:
Step-by-step explanation:
Answer:
[tex]\begin{array}{|c|c|c|c|}\cline{1-4}\vphantom{\dfrac12}\sf Location&AB&BC&AC\\\cline{1-4}\vphantom{\dfrac12} A(3,4),\;B(1,1),\;C(5,1)&3.61&4&3.61\\\cline{1-4}\vphantom{\dfrac12} A(4,5),\;B(2,1),\;C(7,3)&4.47&5.39&3.61\\\cline{1-4}\end{array}[/tex]
[tex]\begin{array}{|c|c|c|c|}\cline{1-4}\vphantom{\dfrac12}\sf Location&m \angle A&m \angle B&m \angle C\\\cline{1-4}\vphantom{\dfrac12} A(3,4),\;B(1,1),\;C(5,1)&67.38^{\circ}&56.31^{\circ}&56.31^{\circ}\\\cline{1-4}\vphantom{\dfrac12} A(4,5),\;B(2,1),\;C(7,3)&82.87^{\circ}&41.63^{\circ}&55.49^{\circ}\\\cline{1-4}\end{array}[/tex]
Step-by-step explanation:
Step 1Place points A, B and C on the coordinate grid.
Alternatively, type the following into the input field as 3 separate inputs:
Triangle 1
A = (3, 4)B = (1, 1)C = (5, 1)Triangle 2
A = (4, 5)B = (2, 1)C = (7, 3)Step 2Use the Segment tool to join each pair of points.
Alternatively, type Segment( <Point>, <Point> ) into the input field (replacing <Point> with the letter name of the point) to create a segment between two points.
Record the length of each side.
[tex]\begin{array}{|c|c|c|c|}\cline{1-4}\vphantom{\dfrac12}\sf Location&AB&BC&AC\\\cline{1-4}\vphantom{\dfrac12} A(3,4),\;B(1,1),\;C(5,1)&3.61&4&3.61\\\cline{1-4}\vphantom{\dfrac12} A(4,5),\;B(2,1),\;C(7,3)&4.47&5.39&3.61\\\cline{1-4}\end{array}[/tex]
Step 3Use the Angle tool to measure each angle in the resulting triangle.
Alternatively, type Angle(Polygon(A, B, C)) into the input field to create all interior angles.
Record the measure of each angle.
[tex]\begin{array}{|c|c|c|c|}\cline{1-4}\vphantom{\dfrac12}\sf Location&m \angle A&m \angle B&m \angle C\\\cline{1-4}\vphantom{\dfrac12} A(3,4),\;B(1,1),\;C(5,1)&67.38^{\circ}&56.31^{\circ}&56.31^{\circ}\\\cline{1-4}\vphantom{\dfrac12} A(4,5),\;B(2,1),\;C(7,3)&82.87^{\circ}&41.63^{\circ}&55.49^{\circ}\\\cline{1-4}\end{array}[/tex]
Note: All measurements have been given to the nearest hundredth (2 decimal places).
A store purchased a stylus for $22.00 and sold it to a customer for 20% more than the purchase price. The customer was charged a 6% tax when the stylus was sold. What was the customer’s total cost for the stylus?
Answer: $27.98
Step-by-step explanation:
22.00 × .2= 4.40
22 + 4.40 = 26.40
26.40 × .06 = 1.584
26.40 + 1.584 = 27.984
Round to the nearest hundred so the total paid by the customer would be 27.98
The coordinates of three vertices of a parallelogram are (3,-2), (5,2) and (0,2). What are the coordinates of the fourth vertex?
The coordinate of fourth point of a parallelogram is (2,0)
We need to remember that the diagonals of a parallelogram intersect each other at a halfway point and the midpoint of each diagonal is the same.
A parallelogram is a geometric object with sides that are parallel to one another in two dimensions. It is a form of polygon with four sides (sometimes known as a quadrilateral) in which each parallel pair of sides have the same length. A parallelogram's adjacent angles add up to 180 degrees.
The midpoint formula
[tex]M=\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}[/tex]
AC=BD
We can find the coordinates of the fourth vertex (x,y) through this procedure:
For x
[tex]\frac{5-2}{2}=\frac{0+x}{2}\\\\x=2\\\\for\ y \\\frac{2-2}{2}=\frac{y+2}{2}\\\\y=0[/tex]
Hence the coordinate of fourth point is (2,0)
learn more about The midpoint formula ,
https://brainly.com/question/13200165
#SPJ4
You are buying 30 acres of farm land at $12000 per acre. What is total cost?
360000 acres
Step-by-step explanation:
12000 x 30= 360000
In △ △ ABC, CJ = 18. If CG = BG, what is KJ? Triangle A B C is divided by 4 segments. A H is the height. C J extends from C to side A B. B I extends from B to side A C. H I extends from the height on B C to I on A C. C J and B I intersect at point K. A J and B J are congruent. A I and C I are congruent.
Solving for CI in terms of the given lengths, we get: [tex]Cl=\frac{\sqrt{BG^{2} -IM^{2} } }{\sqrt{2} }[/tex]
Substituting this expression for CI and the given value for CG into the expression for BI, we get: [tex]BI=CG-\frac{\sqrt{BG^{2} -IM^{2} } }{\sqrt{2} }[/tex].
What is triangle?A triangle is a three-sided polygon, which is a closed two-dimensional shape with straight sides. In a triangle, the three sides connect three vertices, or corners, and the angles formed by these sides are called the interior angles of the triangle. The sum of the interior angles of a triangle is always 180 degrees. Triangles can be classified by their side lengths and angle measurements. For example, an equilateral triangle has three sides of equal length, and all of its angles are 60 degrees; an isosceles triangle has two sides of equal length, and its base angles are also equal; a scalene triangle has three sides of different lengths, and all of its angles are also different. Triangles are a fundamental shape in mathematics and geometry, and they have numerous applications in fields such as architecture, engineering, physics, and more.
Given by the question.
Based on the given information, we can start by drawing a diagram of triangle ABC and the segments AH, BJ, CI, CJ, and BI as described.
Since CG = BG, we can draw the perpendicular bisector of side AC passing through point G, which will intersect side AB at its midpoint M.
Now, we can see that triangle CGB is isosceles with CG = BG, so the perpendicular bisector of side CB also passes through point G. This means that G is the circumcenter of triangle ABC, and therefore, the distance from G to any vertex of the triangle is equal to the radius of the circumcircle.
Next, we can use the fact that AJ and BJ are congruent to draw the altitude from point J to side AB, which we will call JN. Similarly, we can draw the altitude from point I to side BC, which we will call IM.
Since AJ and BJ are congruent, the altitude JN will also be the perpendicular bisector of side AB, so it will pass through point M. Similarly, the altitude IM will pass through point G, which is the circumcenter of triangle ABC.
Now, we can use the Pythagorean theorem to find the lengths of JN and IM in terms of the given lengths:
[tex]JN^{2}= AJ^{2} -AN^{2} \\ = ( AH+HN)^{2} - AN^{2} \\=AH^{2} +2AH*HN+HN^{2}-AN^{2} \\[/tex]
[tex]IM^{2}= CI^{2} -CM^{2} \\=( CG-GM)^{2} -CM^{2} \\CG^{2}-2CG*GM+GM^{2} -CM^{2}[/tex]
Since CG = BG and GM = BM (since M is the midpoint of AB), we can simplify the expression for IM^2 as follows:
[tex]IM^{2}[/tex] = [tex]BG^{2}[/tex] - 2BG * BM + [tex]BM^{2}[/tex] - [tex]CM^{2}[/tex]
= [tex]BG^{2}[/tex] - [tex]BM^{2}[/tex] - [tex]CM^{2}[/tex]
Now, we can use the fact that BJ and CI intersect at point K to find the length of KJ:
KJ = BJ - BJ * (CK/CI)
= BJ * (1 - CK/CI)
= BJ * (1 - BM/CM)
To find BM/CM, we can use the fact that triangle BCI is isosceles with BI = CI, so the altitude IM is also a median of the triangle. This means that CM = 2/3 * BI. Similarly, we can find BJ in terms of JN using the fact that triangle ABJ is isosceles with AJ = BJ:
BJ = 2 * JN
Substituting these expressions into the equation for KJ, we get:
KJ = 2 * JN * (1 - 2/3 * BI/CM)
Now, we just need to find BI/CM in terms of the given lengths. Using the fact that triangle BCI is isosceles with BI = CI, we can find BI in terms of CG:
BI = CG - CI
Substituting this expression into the equation for [tex]IM^{2}[/tex]and simplifying, we get:
[tex]IM^{2}[/tex] =[tex]BG^{2}[/tex] - CG * CI
To learn more about Pythagorean theorem:
https://brainly.com/question/28361847
#SPJ1
7. Hal records the numbers of winners
of a contest in which the player
chooses a marble from a bag.
DA G
Game 1
Game 2
Game 3
Number of Number of
Players
Winners
123
52
155
63
172
65
Based on the data for all three
games, what is the experimental
probability of winning the contest?
Express the answer as a decimal.
Answer:0.40
Step-by-step explanation:
the experimental probability of winning the contest is 0.4 or 40%.
Define probabilityProbability is a measure of the likelihood or chance of an event occurring. It is a number between 0 and 1, where 0 denotes the impossibility of the occurrence and 1 denotes its certainty. An occurrence is more likely to occur the higher its probability.
Players: 123 + 155 + 172, or 450 total
52 plus 63 plus 65 winners make up the total of 180.
Experimental probability of winning the contest = Total number of winners / Total number of players
Experimental probability of winning the contest = 180 / 450 = 0.4
Hence, the experimental probability of winning the contest is 0.4 or 40%
To know more about occurrence, visit:
https://brainly.com/question/28953812
#SPJ1