in the coordinate plane, the vertices of triangle PAT are P(-1,-6), A(-4,5), and T(5,-2). Prove that PAT is an isosceles triangle.
In the coordinate plane, the vertices of triangle PAT are P(-1,-6), A(-4,5), and T(5,-2) forms an isosceles triangle.
To prove that triangle PAT is an isosceles triangle, we need to show that at least two sides of the triangle are congruent.
Let's calculate the distances between the vertices:
Distance between P and A:
PA = √[(x₂ - x₁)² + (y₂ - y₁)²]
= √[(-4 - (-1))² + (5 - (-6))²]
= √[9 + 121]
= √130
Distance between P and T:
PT = √[(5 - (-1))² + (-2 - (-6))²]
= √[(5 + 1)² + (-2 + 6)²]
= √[36 + 16]
= √52
= 2√13
Distance between A and T:
AT = √[(5 - (-4))² + (-2 - 5)²]
= √[(5 + 4)² + (-2 - 5)²]
= √130
We can see that PA = AT = √130, and PT = 2√13.
Since PA = AT, we have shown that two sides of the triangle are congruent.
Therefore, triangle PAT is an isosceles triangle.
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Use the drawing tools to form the correct answer on the provided grid.
Triangle ABC undergoes a sequence of transformations.
1. Reflection across the line x = 1.
2. Dilation by a factor of 2 centered at the origin.
What is the resulting image?
Note that the resulting image where the above transformation takes place is attached accordingly.
How did we arrive at this?Current coordinates given from the image
A( 2, 7)
B (3, 2)
C (0, 2)
1) To reflect a point across the line x = 1, we need to find the distance between the point and the line and then move it to the same distance on the other side of the line.
In this case, all points will have their x-coordinate mirrored with respect to the line x = 1.
New coordinates after reflection is
A'(0, 7)
B'(−1, 2)
C'(2, 2)
2 - Dilation by a factor of 2 centered at the origin:
To dilate a point by a factor of 2 centered at the origin, we need to multiply both the x and y coordinates of the point by 2.
New coordinates after dilation is
A''(0, 14)
B''(−2, 4)
C''(4, 4)
Plotting the above will given the resultant image int he attached.
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Find the measure of each specified angle or arc
Arc JK
Angle JHI
Arc IJL
Arc JKL
H
30°
K
L
180
90
60
120
Answer:
the answer is 60
Step-by-step explanation:
rewrite each of the following expressions without using absolute value:
∣4r-12∣if r<3
Answer:
12-4r
Step-by-step explanation:
4r<12 because r<3 so you flip it.
Suppose $6000 is invested at 3% interest compounded continuously. How long will it take for the investment to grow to $12000?
The time it takes for the investment to grow to $12,000 if the interest is compounded continuously is 23.1 years.
Given that,
Principal amount invested, P = $6000
Rate of interest, r = 3% = 3/100 = 0.03
If the interest compounded continuously,
Final amount, A = P e^(rt)
12000 = 6000 e ^(0.03t)
2 = e ^(0.03t)
Taking logarithms on both sides,
ln (2) = 0.03t
t = 23.1 year.
Hence the time is 23.1 years.
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find the exact value of Sin A
Step-by-step explanation:
sin A = opposite/ Hypotenuse
Sin A = 5/7
Hello !
sin(A) = opposite/hypotenuse = 5/7
arcsin(5/7) ≈ 45,58°
sin(A) = 5/7
the angle A ≈ 45,58°
write and solve an euation to find the measure of angle x
Answer:
a= 65, b=75
Step-by-step explanation:
Angle a= 180-70-45
=65
Angle b= 180-60-45
=75
The half life of Radium is 1620 years. When will 20 g sample ony have 15 g left?
Answer:
The formula for radioactive decay is:
N = N₀ * (1/2)^(t/T)
where:
N₀ = initial amount
N = remaining amount
t = time elapsed
T = half-life
Let's plug in the given values:
N₀ = 20 g
N = 15 g
T = 1620 years
15 = 20 * (1/2)^(t/1620)
Dividing both sides by 20:
0.75 = (1/2)^(t/1620)
Taking the logarithm base 1/2 of both sides:
log(0.75) = t/1620 * log(1/2)
Solving for t:
t = log(0.75) / log(1/2) * 1620
t ≈ 623 years
Therefore, it will take approximately 623 years for a 20 g sample of Radium to decay to 15 g.
Step-by-step explanation:
28. Which unit rate is equivalent to the ratio 12 to 3?
A. 12/1
B. 4/1
C. 9/1
D.3/1
Answer:
B. 4/1
Step-by-step explanation:
12/3 can be divided by 3.
Therefore 4/1.
Help what is the answer
Answer:
175 cm²
Step-by-step explanation:
Calculate each shape separately, then add them together.
Parallelogram: A = bh = (14)(5) = 70
Trapezium: A = ((a + b)/2)(h) = ((21 + 14)/2)(11-5) = 105
Total area = 70 + 105 = 175 cm²
[1] 2 (a) Using the information given in the advertisement shown, find the sale price of the table. Answer.... SALE All prices reduced by 30% Save $180 on this table
Based on the information given in the advertisement, we can conclude that the sale price of the table is $420.
Here's how we can arrive at this answer:
Let the original price of the table be represented by P.
We know that the sale price of the table is obtained by reducing the original price by 30%. Mathematically, this can be represented as:
Sale price = P - 0.3P
Simplifying this expression, we get:
Sale price = 0.7P
We are also given that the sale saves us $180 on this table. Mathematically, this can be represented as:
0.7P - P = -$180
Simplifying this expression, we get:
-0.3P = -$180
Dividing both sides by -0.3, we get:
P = $600
Therefore, the original price of the table was $600.
Using the equation for sale price that we derived earlier, we can now find the sale price of the table:
Sale price = 0.7P = 0.7 x $600 = $420
Hence, the sale price of the table is $420.
I don’t know what this is trying to tell can someone make it more easier to understand
Please find attached the graph of the straight line equation; y = -2·x + 3/2, which is the same as the graph of the equation; 4·x + 2·y = 3, created with MS Excel
What is an equation of a straight line graph?An equation of a straight line is an equation of the form; y = m·x + c
b. 1. The equation of the line is; 4·x + 2·y = 3
The above equation of the line can be expressed in slope-intercept form; y = m·x + c, where; m is the slope of the graph of the equation and c is the y-coordinate of the y-intercept, by making y the subject as follows;
2·y = 3 - 4·x
y = (3 - 4·x)/2
y = 3/2 - 2·x
Therefore; y = -2·x + 3/2
The above equation indicates;
The slope, m = -2
The y-intercept, c = 3/2
The coordinate of the y-intercept is therefore; (0, 3/2)
The point (0, 3/2) is to be plotted on the graph
2. In order to draw the graph, the coordinates of a second point can be found as follows;
The slope of the graph = The ratio of the rise to the run of the graph
Slope = Rise/Run = Δy/Δx
The rise = The number of units a point progresses vertically
The run = The number of units a point progresses horizontally
Therefore, a slope of -2, indicates;
Slope = Rise/run = Δy/Δx = -2 = -2/1
A rise of -2 units is accompanied by a run of 1 unit
In terms of x and y an increase of 1 unit in the x-value is accompanied in the y-value by a -2 units increase.
Therefore, a second point on the graph is; (0 + 1, ((3/2) - 2)) = (1, -1/2)
3. The line of the graph therefore passes through both (0, 3/2) and (1, -1/2)
Drawing a line through the above two points creates the graph of the equation, 4·x + 2·y = 3
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Determine the values of max{3,π, √2}
The maximum value among 3, π, and √2 is π (approximately 3.14159).
To determine the maximum value among 3, π, and √2, we compare the numbers to find the largest one.
Comparing 3 and π:
Since π (approximately 3.14159) is greater than 3, we can eliminate 3 from consideration.
Comparing π and √2:
To compare these two numbers, we can square both of them. Squaring π yields approximately 9.8696, while squaring √2 gives us exactly 2. Since 9.8696 is greater than 2, we can eliminate √2 from consideration.
Thus, the maximum value among 3, π, and √2 is π (approximately 3.14159).
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In the following diagram,
What is the measure of
∠x
Answer:
Step-by-step explanation:
BAD = 42
x + 103 + 42 = 180
x = 180 - 145
x = 35
The diagram shows a scale drawing of the
side elevation of a building.
3 cm represents 1 m.
What is the width of the building in metres?
(Give your answer in meters).
Answer:
The answer for the width of the building is 5m
Step-by-step Explanation:
3cm=1m
15cm=x
cross multiply
x×3=15×1
3x=15
divide both sides by 3
3x/3=15/3
x=5m
Show that the path of a moving point parallel to the axes of x and y with velocitiesu +
ey andv + ex is a conic section
We have shown that the path of a moving point with velocities u + ey and v + ex, parallel to the axes of x and y, is either a line (when v - [tex]e^2[/tex] ≠ 0) or a horizontal line (when v - [tex]e^2[/tex] = 0), both of which are conic sections.
To show that the path of a moving point parallel to the axes of x and y with velocities u + ey and v + ex is a conic section, we can analyze the motion of the point using the principles of calculus and conic sections.
Let's denote the position of the point at any given time t as (x, y). We are given that the velocities along the x and y axes are u + ey and v + ex, respectively. This means that the derivatives of x and y with respect to time, dx/dt and dy/dt, can be expressed as:
dx/dt = u + ey
dy/dt = v + ex
Now, let's integrate these expressions to obtain x and y as functions of t. Integrating dx/dt with respect to t gives:
x = ut + eyt + C1
Similarly, integrating dy/dt with respect to t gives:
y = vt + ext + C2
Where C1 and C2 are constants of integration.
Now, we can eliminate the parameter t by expressing t in terms of x and y. From the equation y = vt + ext + C2, we can solve for t:
t = (y - ext - C2) / v
Substituting this value of t into the equation for x, we get:
x = u[(y - ext - C2) / v] + ey[(y - ext - C2) / v] + C1
Simplifying this equation, we obtain:
vx - u - evx + ue + vy - [tex]e^2[/tex]x - eyC2 = C1v
Rearranging the terms, we get:
vx - vy + ue + evx - [tex]e^2[/tex]x = C1v + eyC2 - u
Let's define new constants A = ue + ev and B = C1v + eyC2 - u. The equation then becomes:
(v - [tex]e^2[/tex])x + (u + ev)y = A + B
This equation is in the standard form of a conic section, specifically a line. However, we can manipulate this equation further to reveal other possible conic sections.
Let's consider the case when v - [tex]e^2[/tex] ≠ 0. In this case, we can divide both sides of the equation by v - [tex]e^2[/tex], yielding:
x + [(u + ev)/(v - [tex]e^2[/tex])]y = (A + B)/(v - [tex]e^2[/tex])
Now, let's define another constant C = (u + ev)/(v -[tex]e^2[/tex]) and rewrite the equation as:
x + Cy = D
Where D = (A + B)/(v - [tex]e^2[/tex]).
This equation represents a line in the x-y plane.
On the other hand, if v - [tex]e^2[/tex] = 0, the equation becomes:
0x + (u + ev)y = A + B
This simplifies to:
(u + ev)y = A + B
Which is a horizontal line parallel to the x-axis.
Therefore, we have shown that the path of a moving point with velocities u + ey and v + ex, parallel to the axes of x and y, is either a line (when v - [tex]e^2[/tex] ≠ 0) or a horizontal line (when v - [tex]e^2[/tex] = 0), both of which are conic sections.
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Determine the equation of the circle graph below
The equation of the circle graphed is (x + 6)² + y² = 4
How to determine the equation of the circle graphedFrom the question, we have the following parameters that can be used in our computation:
The circle
Where, we have
Center = (a, b) = (-6, 0)
Radius, r = 2 units
The equation of the circle graphed is represented as
(x - a)² + (y - b)² = r²
So, we have
(x + 6)² + y² = 2²
Evaluate
(x + 6)² + y² = 4
Hence, the equation is (x + 6)² + y² = 4
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Solve the system of equations below using substitution.
y= 6x - 11
2x + 3y = 7
What's the solution of the system?
O A.(-2.-1)
O B.(-1,-2)
O C. (2,1)
O D. (1.2)
The correct option is C. (2, 1).
Given equations:
y = 6x - 11
2x + 3y = 7
We'll start by solving equation 1) for y:
y = 6x - 11
Now, substitute this value of y into equation 2):
2x + 3(6x - 11) = 7
Simplify the equation:
2x + 18x - 33 = 7
20x - 33 = 7
Add 33 to both sides:
20x = 40
Divide both sides by 20:
x = 2
Now, substitute the value of x back into equation 1) to find y:
y = 6(2) - 11
y = 12 - 11
y = 1
Therefore, the solution to the system of equations is (x, y) = (2, 1).
What is the area of a triangle
Answer:A=BxH divided by 2
_
Step-by-step explanation:
Find the missing parts that would make this equation true.
3, 2x(x-2y) + y(x-2y)-(x-2y) = (2x +_-_(-2y)
The missing parts that would make the equation 2x(x-2y) + y(x-2y)-(x-2y) = (2x +_ - _ -2y) true are ² - x - 3xy - 2y²
How to determine the missing parts that would make the equation true.From the question, we have the following parameters that can be used in our computation:
2x(x-2y) + y(x-2y)-(x-2y) = (2x +_ - _ -2y)
When expanded, we have
2x² - 4xy + xy - 2y² - (x - 2y) = (2x +_ - _ -2y)
Open the brackets
So, we have
2x² - 4xy + xy - 2y² - x - 2y = (2x +_ - _ -2y)
Evaluate the like terms
2x² - 3xy - 2y² - x - 2y = (2x +_ - _ -2y)
Rewrite as
2x² - x - 3xy - 2y² - 2y = (2x +_ - _ -2y)
This means that the missing parts are ² - x - 3xy - 2y²
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Simone has 16 square inch tiles she glues all of them on cardboard to make two different rectangles each with the same area what are the side lengths of the two rectangles she can make
Simone can make two different rectangles with the following side lengths:
Rectangle 1: Length = 16 inches, Width = 1 inch
Rectangle 2: Length = 8 inches, Width = 2 inches
To find the side lengths of the two rectangles that Simone can make using 16 square inch tiles, we need to consider the factors of 16 and check which pairs of factors can form rectangles with the same area.
The factors of 16 are 1, 2, 4, 8, and 16. We can pair these factors to form different rectangles and calculate their areas:
Pairing: 1 and 16
Length: 16
Width: 1
Area: 16 square inches
Pairing: 2 and 8
Length: 8
Width: 2
Area: 16 square inches
Pairing: 4 and 4
Length: 4
Width: 4
Area: 16 square inches
Therefore, Simone can make two different rectangles with the following side lengths:
Rectangle 1: Length = 16 inches, Width = 1 inch
Rectangle 2: Length = 8 inches, Width = 2 inches
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complete the fraction that is equivalent to 3/16.
Answer:
[tex]\frac{9}{48}[/tex]
Step-by-step explanation:
the 3 on the numerator has been multiplied by 3 to get the 9 on the numerator on the right.
the same operation must be applied to the 16 on the denominator of the fraction.
16 × 3 = 48
then
[tex]\frac{3}{16}[/tex] = [tex]\frac{3(3)}{16(3)}[/tex] = [tex]\frac{9}{48}[/tex]
Spiderman ascends a building to its peak The peak is 812ft above sea level. Spiderman then descends 30ft to face the Chameleon. Find the Spiderman's evevation above the sea level after meeting the Chameleon.
A. 812 ft above sea level
B. 30 ft below sea level
C. 782 ft above sea level
Spiderman's elevation above sea level after meeting the Chameleon is 782ft.
An elevation is the view of a 3D shape when it is looked at from the side or from the front.
Angle of elevation: Angle of elevation is the angle between the horizontal line and the line of sight. It is formed at the vertex of intersection of the horizontal line and line of sight. It is the same angle as used in trigonometry.
To calculate Spiderman's elevation above sea level after descending 30ft to meet the Chameleon, we need to subtract 30ft from the initial elevation of 812ft.
812ft - 30ft = 782ft
Therefore, Spiderman's elevation above sea level after meeting the Chameleon is 782ft.
So, the correct answer is C. 782ft above sea level.
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How do I do the problem correctly? please explain i will mark you brainliest !!
Answer:
Explanation: Since these two powers have the same base of 5, you can multiply them together by simply adding their exponents to get
5^11
A common mistake in this problem would be to multiply
the bases together and give an answer of
25^11
.
The 5's in this problem can not be multiplied together because they are not coefficients. They are bases.
When applying your exponent rules, your base will never change.
What is the volume of a sphere with a radius of 25.9 in, rounded to the nearest tenth of a cubic inch?
I want to solve for x
Answer:
x = [tex]\frac{A*t^2}{800}[/tex]
Step-by-step explanation:
First move t² to the left side by multiplying it.
A×t² = 800x - to get rid of 800, we divide both sides by 800. This isolates x.
x = [tex]\frac{A*t^2}{800}[/tex]
Help pls What is the general solution to the trigonometric equation
The general solution to the differential equation -√3 cscθ = 2 is θ = 2π/3 + nπ.
To find the general solution to the trigonometric equation -√3 cscθ = 2, where n is an integer, we need to solve for θ.
Let's start by rewriting the equation:
-√3 cscθ = 2
Since cscθ is the reciprocal of sinθ, we can rewrite the equation as:
-√3 / sinθ = 2
To solve for θ, we can take the reciprocal of both sides:
sinθ / -√3 = 1/2
Next, we can take the reciprocal of both sides again:
-√3 / sinθ = 2
Now, we can find the values of θ that satisfy this equation. The general solution is given by:
θ = arcsin(-√3/2) + nπ
θ = 2π/3 + nπ
where n is an integer.
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Explain how you know that (3, 5) is not a solution to the given inequality by looking at the graph.
The reason that the point (3, 5) is not a solution to the given inequality is given below.
We are given that;
y > 2x + 1
Now,
The inequality y > 2x + 1 can be graphed by first graphing the boundary line y = 2x + 1. Since the inequality is strict (y >), we draw a dashed line to indicate that points on the line are not solutions to the inequality. Then, we shade the region above the line to indicate all points that satisfy the inequality.
If we have (3,5) as a point, we can see that it lies on the boundary line
y = 2x + 1.
Since the inequality is strict (y >), points on this line are not solutions to the inequality
Therefore, by the inequality the answer will be given above.
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The equation to the graph shown is y = ax + p, where a and p are real numbers. What
is true about a and p?
The true statement about a and p is a is positive and p is positive.
We have the equation
y= ax+ p
where a and p are real numbers.
In general, "a" represents the slope of the line (the rate of change of y with respect to x).
and, "p" represents the y-intercept (the value of y when x = 0).
The specific values of a and p can be determined by examining the graph or given data points.
By looking the graph, the slope is positive
b is the y intercept, where x = 0 and that is positive.
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5. (a) There are 160 apples and 224 mangoes in a bag. i. ii. What is the greatest number of children to distribute these apples and mangoes equally? How many fruits of each kind will each child get?
Hello !
You find the GCD of 160 : 224
160
= 2 x 80
= 2 x 2 x 40
= 2 x 2 x 2 x 20
= 2 x 2 x 2 x 2 x 10
= 2 x 2 x 2 x 2 x 2 x 5
224
= 2 x 112
= 2 x 2 x 56
= 2 x 2 x 2 x 28
= 2 x 2 x 2 x 2 x 14
= 2 x 2 x 2 x 2 x 2 x 7
GDC = 2 x 2 x 2 x 2 x 2 = 32
=> 32 childrens
160/32 = 5
224/32 = 7
=> 5 apples and 7 mangoes per child