Circle P is tangent to the x-axis and the y-axis. If the coordinates of the center are (r, r), find the length of the chord whose endpoints are the points of tangency.
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Answer: The length of the chord AB is √2 r units.
Step-by-step explanation: Given in the question that Circle P is tangent to the X-axis at the point A and tangent to the Y-axis at the point B.
Also, the co-ordinates of centre of the circle P are C(r, r). See the attached image below.
We are to find the length of the chord AB, with end-points A and B as the points of tangency.
From the figure, we note that
the co-ordinates of the points A and B are (0, r) and (r, 0) respectively.
As the X-axis and Y-axis are perpendicular to each other, so the lines parallel to them, OB and OA are also perpendicular to each other.
That is, ∠ACB = 90°.
This implies that the triangle ACB is a right-angled triangle.
Now, using Pythagoras theorem in ΔACB, we have
[tex]AB^2=CA^2+CB^2~~~~~~~~~~~~~~~~~~~~~(i).[/tex]
The lengths of the line segments CA and CB are calculated, using distance formula, as follows:
[tex]CA=\sqrt{(0-r)^2+(r-r)^2}=\sqrt{r^2}=r,\\\\CB=\sqrt{(r-r)^2+(0-r)^2}=\sqrt{r^2}=r.[/tex]
From equation (i), we get
[tex]AB^2=r^2+r^2\\\\\Rightrarrow AB^2=2r^2\\\\\Rightarrow AB=\sqrt2r~\textup{units}.[/tex]
Thus, the length of the chord AB is √2 r units.
A circle, tangent to the x-axis and y-axis, forms a square where its radius r is the side length. The chord, which is the diagonal, can be found using the Pythagorean theorem resulting to r√2.
Explanation:In the given problem, a circle P is tangent to the x-axis and y-axis at two points. This means that the radius of the circle is the distance from the center to either axis, which are both r, as the coordinates of the center are (r,r). We are asked to find the length of the chord whose endpoints are the points of tangency.
The chord is actually the diagonal of a square that has sides of length r. So, its length can be found using the Pythagorean theorem (a^2 + b^2 = c^2), where a and b are the lengths of the sides of the square and c is the diagonal. Hence the length of the chord can be found as √(r^2 + r^2) = √(2r^2) = r√2.
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Related Questions
I need help ASAP! Anyone help?
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A coin is tossed 5 times how many different outcomes are possible
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Paula is working on her English essay. She already has 150 words written and has planned out the rest. Paula's essay must be 3,000 words in length, and she types at a rate of 38 words per minute.
Which equation represents the number of words in her essay, and how many minutes, m, will it take Paula to complete it?
a) 38m - 150 = 3,000; m = 75 minutes
b) 38m - 150 = 3,000; m = 83 minutes
c) 38m + 150 = 3,000; m = 75 minutes
d) 38m + 150 = 3,000; m = 83 minutes
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The correct equation for the given scenario is 38m + 150 = 3,000. Solving this equation, we find that it will take Paula approximately 75 minutes to finish her essay.
Explanation:The question involves calculating how much time Paula needs to complete her essay based on her typing speed and desired essay length. Paula needs to write 3,000 words, but she already has 150 written. Therefore, she needs to write 3,000 - 150 = 2,850 more words. If she types at a rate of 38 words per minute, the number of minutes, represented as m, can be found using the equation 38m = 2,850. Therefore, the correct equation from the options is 38m + 150 = 3,000. Solving this equation for m gives m = 2,850 / 38 ≈ 75 minutes. Thus, option c) 38m + 150 = 3,000; m = 75 minutes is the correct answer.
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Find two numbers differing by 44 whose product is as small as possible.
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Given that the product of two numbers will be smallest when the two numbers are equal, the numbers closest to this rule given the condition of having a fixed difference of 44 are -22 and 22.
Explanation:This problem falls under the branch of Mathematics called optimization problems in calculus. In this case, we are asked to find two numbers differing by 44 whose product is smallest. Remember that the product of two numbers that are a fixed difference apart will be smallest when the two numbers are equal. But since we can't have that according to the problem's conditions, we will have to get the closest possible which is making the two numbers as close to each other as possible.
Let's denote the smaller number as 'x'. Consequently, the larger number will be 'x + 44'. The product of these two numbers (which we want to minimize) is given by the function f(x) = x * (x + 44). To find the minimum, you would typically take the derivative of this function, set it equal to zero, and solve for 'x'. However, since we're dealing with a quadratic function, we know the graph is a parabola that opens upwards, and thus the minimum point is at the vertex. In a quadratic function given in the form of f(x) = a*(x - h)² + k, the vertex is at the point (h, k). Comparing this with our quadratic function, we can see by inspection that our 'h' is '-22'. Therefore, the two numbers are -22 and 22.
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The smallest product of two numbers differing by 44 is -484. These numbers are -22 and 22.
Explanation:This problem is solvable using the principle of quadratic minimum. Let's denote the two numbers as x and x + 44. Their product equals x(x + 44) = x2 + 44x.
This is a quadratic function with a positive leading coefficient (x2), therefore, it will yield a minimum value. The x-value for this minimum can be found using the formula -b/2a for any quadratic ax2 + bx + c, where a = 1 and b = 44. Calculating, -b/2a = -44/2 = -22.
This means that the product will be smallest when x = -22. However, since we're looking for two numbers that differ by 44, the necessary pairs are -22 and 22. The product of these two numbers, (-22)(22), equals -484, which is indeed the smallest possible product for two numbers differing by 44.
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Which statements are correct interpretations of this graph?
Select each correct answer.
1 2/3 of a page is edited per minute
3 pages are edited every 5 min
6/10 of a page is edited per minute
5 pages are edited every 3 min
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A total of 720 people attended the school-basketball game. adult tickets cost $2.50 each and student tickets cost $1.50 each. if $1220 worth of tickets were sold, how many students and how many adults attended?
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Which two variables are related in bernoulli's equation?
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which equation represents a linear function? equation 1: y = 2x2 1 equation 2: y2 = 3x 1 equation 3: y = 5x – 1 equation 4: y = 4x4 – 1 (5 points) equation 1 equation 2 equation 3 equation 4
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linear means first degree
first degree means the hightest exponent of the variables is 1
so
equation 1:
[tex]y=2x^2+1[/tex]
highest exponent is 2, not 1
not leinear
equation 2: y^2=3x+1
highest exponent is 2, not 1
not linear
equaton 3 y=5x-1
y=5x^1-1, it is 1
it is linear
equaton 4
y=4x^4-1
highest is 4, not 1
not linear
answer is equation 3: y=5x-1
Answer:
equation 3
Step-by-step explanation:
What is 10 more than 451
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HELP ASAP!!!!!!!!!!!!
Which property justifies the following statement? ∠AOB ≅ ∠AOB
Transitive Property of Congruence
Reflexive Property of Congruence
Symmetric Property of Congruence
Distributive Property
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Reflexive Property of Congruence justifies the following statement ∠AOB ≅ ∠AOB. Option B is correct.
What are the angle?Orientation of the one line with respect to the horizontal or other respective line is known as a measure of orientation and this measure is known as the angle.
Here,
∠AOB ≅ ∠BOB
It can be obtained, only by when the angle gets reflected.
So by the reflexive Property of Congruence justified.
Thus, the reflexive Property of Congruence justifies the following statement ∠AOB ≅ ∠AOB. Option B is correct.
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20 points!
Show work please! I have no idea how to do this, haha.
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[tex]\bf 3^{5x-9}=\left( \cfrac{1}{27} \right)^{2x-8}\qquad \boxed{27=3^3}\qquad 3^{5x-9}=\left( \cfrac{1}{3^3} \right)^{2x-8} \\\\\\ 3^{5x-9}=\left( 3^{-3} \right)^{2x-8}\implies 3^{5x-9}= 3^{-3(2x-8)} \\\\\\ 3^{5x-9}= 3^{-6x+24}\impliedby \begin{array}{llll} \textit{the bases are the same, then}\\ \textit{the exponents must be the same} \end{array} \\\\\\ 5x-9=-6x+24\implies 11x=33\implies x=\cfrac{33}{11}\implies x=3[/tex]
Rearrange the formula c2 = a2 + b2 for b.
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Answer:
[tex]b = \sqrt{csquare - a square[/tex]
Step-by-step explanation:
Reiko started a business selling home medical supplies. She spent $6500 to obtain her merchandise, and it costs her $550 per week for general expenses. She earned $900 per week in sales. What is the minimum number of weeks it will take for Reiko to make a profit? Write an inequality to model the problem.
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Find limit as x approaches 4 of the quotient of the quantity the square root of the quantity 3 times x plus 4 minus the square root of 4 times x and the quantity x squared minus 4 times x . you must show your work or explain your work in words
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multiply by the conjugate (√3x+4 - 2√x)(√3x+4 + 2√x) / (x^2-4x) (√3x+4 + 2√x)
simplify = -x+4 / (x^2-4x) (√3x+4 + 2√x)
factor (x^2-4x) which equals x(x-4)
-x+4 / x(x-4) (√3x+4 + 2√x)
simplify= -1/ 4(√3x+4 + 2√x)
next plug in 4
-1/ 4(√3(4)+4 +2 √(4)) = (-1/32)
BEST ANSWER GETS BRAINLIEST PLEASE HELP
Answers
11. y - y₁ = m(x - x₁)
y - 4 = ²/₃(x - 3)
y - 4 = ²/₃x - 2
y = ²/₃x + 2
12. y - y₁ = m(x - x₁)
y - 0 = ⁻¹/₅(x - 5)
y - 0 = ⁻¹/₅x + 1
y = ⁻¹/₅x + 1
The perimeter of the triangle below is 55 units. find the value of z.
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The area of the parallelogram is (4x+16)(4x+16) square feet. write an expression for the base. expression for the base: 16x ft
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Given that the parallelogram has an area of 4x + 16,
i.e. base * height = 4x + 16.
Therefore, an expression for the base of the parallelogram is given by
[tex]Base= \frac{4x+16}{height} \ ft [/tex]
Answer:
[tex]base=\frac{4x+16}{height}[/tex] which is the required expression.
Step-by-step explanation:
We have been given the area of parallelogram which is (4x+16)
And the formula for area of parallelogram is [tex]base\cdot height[/tex]
Hence, when we substitute the value if area of parallelogram in the formula we get:
[tex]4x+16=base\cdot height[/tex]
Now, rearranging the terms so, as to get the expression for the base or value of base in terms of height and area we get:
[tex]base=\frac{4x+16}{height}[/tex] which is the required expression.
A sequence is defined by the recursive function f(n + 1) = –10f(n).
If f(1) = 1, what is f(3)?
Answers
---------------------------------
---------------------------------
Explanation:
First term
f(1) = 1
which is given to us
We will use the equation f(n+1) = -10*f(n) to generate successive terms after this first term.
---------------
Second term
f(n+1) = -10*f(n)
f(1+1) = -10*f(1) ... plug in n = 1
f(2) = -10*f(1)
f(2) = -10*1 ... replace f(1) with 1 (works because f(1) = 1)
f(2) = -10
Put another way, the second term is -10 times the first term
second term = -10*(first term)
second term = -10*(1)
second term = -10
---------------
Third term
f(n+1) = -10*f(n)
f(2+1) = -10*f(2) ... replace n with 2
f(3) = -10*f(2)
f(3) = -10*(-10) ... replace f(2) with -10 (works because f(2) = -10)
f(3) = 100 which is the answer we want
In other words, the third term is -10 times the second term
third term = -10*(second term)
third term = -10*(-10)
third term = 100
---------------
Summary of the three terms
f(1) = 1
f(2) = -10
f(3) = 100
Each term is found by multiplying the previous term by -10
Answer:
100
Step-by-step explanation:
PLEASE HELP WILL BAINLIEST
What is m∠M ?
Enter your answer in the box.
Answers
(3x-16) +(x+6)+x=180
5x-10=180
5x=190
x=38
angle M is 3x-16=3*38-16=98
Two-fifths of a number is decreased by three. Find the value of the expression if the number is twenty-five.
A. 8.8
B. 13
C. 10
D. 7
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0.4 * 25= 10.
(C)10
Answer:
7.
Step-by-step explanation:
2/5x -3
If we plug in 25, we get:
2/5(25) - 3 =
50/5 - 3 =
10 - 3
7
Jack earns $50 for washing cars on the weekend. He makes an average of $4.25 in tips per hour. Write the function of Joe's earnings, and solve for how much he would make after 6 hours.
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Draw two different visual representations of 3/5. Use different shapes for each representation.
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Find the point that is two-thirds of the way from (3,4) to (9,-2)
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[tex]\bf \left. \qquad \right.\textit{internal division of a line segment} \\\\\\ A(3,4)\qquad B(9,-2)\qquad \qquad 2:1 \\\\\\ \cfrac{AC}{CB} = \cfrac{2}{1}\implies \cfrac{A}{B} = \cfrac{2}{1}\implies 1A=2B\implies 1(3,4)=2(9,-2)\\\\ -------------------------------\\\\ { C=\left(\cfrac{\textit{sum of "x" values}}{\textit{sum of ratios}}\quad ,\quad \cfrac{\textit{sum of "y" values}}{\textit{sum of ratios}}\right)}\\\\ -------------------------------[/tex]
[tex]\bf C=\left(\cfrac{(1\cdot 3)+(2\cdot 9)}{2+1}\quad ,\quad \cfrac{(1\cdot 4)+(2\cdot -2)}{2+1}\right) \\\\\\ C=\left( \cfrac{3+18}{3}~~,~~\cfrac{4-4}{3} \right)\implies C=\left( 7~~,~~0 \right)[/tex]
Suppose an isosceles triangle ABC has A = 45° and b = c = 4. What is the length of a2? A.22.63
B.54.63
C.9.37
D.3.10
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Since b = c, the angles B and C also equal
so you get
180 - 45 = 135
135/2 = =67.5 for each angle
now we know 2 sides and all 3 angles, use the law of sines to find a
b/sin B = a/sin A
4/(sin 67.5) = a/(sin 45)
a = 4(sin 45)/(sin 67.5)
a^2 = (4(sin 45)/(sin 67.5))^2
a^2 = 9.37
Answer is C
when ava bought her car, it was worth 19,340. it was expected to decrease at a rate of 12.5% each year. what is the best prediction for value of avas car 10 after she bought it
A.8173
B.6840
C.5088
D.2418
Answers
19340 * (1-0.125)^10 =
19340 * 0.875^10 =
$5087.88 round to 5088
Factor Completely: 54m^5+18m^4+9m^3
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hope this helps
Find the sum and express it in scientific notation. 0.00348 + (6.2 x 10-2)
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0.00348 + 0.062 =
0.06548 =
6.548 * 10^-2 <=
0.00348+(6.2 x 8)
0.00349+(49.6)
=49.60348
In the diagram below, which distance represents the distance from point J to KL
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Answer:
Option B JM is the right answer.
Step-by-step explanation:
The distance that represents the distance from point J to KL is JM.
The distance between any given point and a line is the shortest distance between a fixed point and any point on the line.
This implies that it is the length of the perpendicular line segment on the line and passes through the point.
Here, in the figure, JM is perpendicular to the line KL drawn from the point J.
So, the answer is JM.
Please Help
Simplify.
(y/4)divide(1/4)
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_______________________________________________________
Explanation:
_______________________________________________________
[tex] \frac{y}{4}[/tex] ÷ [tex] \frac{1}{4} [/tex] ;
= [tex] \frac{y}{4}[/tex] * [tex] \frac{4}{1} [/tex] ;
Note: Both of the "4 's " cancel out to "1" ; since "4÷4 = 1" ; and we have:
[tex] \frac{y}{1}[/tex] * [tex] \frac{1}{1} [/tex] ;
= y * 1 = y ;
__________________________________________
The answer is: "y" .
__________________________________________
If Jesse wants to buy a $75,000 10-year term life insurance policy, and the annual premium rate (per $1000 of face value) for his age group is $2.34, how much is Jesse’s annual premium? a. $175.50 b. $3,205.12 c. $234.00 d. $1,755.00
Answers
Jesse’s annual premium would be $175.50. The correct option is A.
What is annual premium?An annual premium is the amount of money paid by an insurance policyholder each year in order to keep their coverage.
It is usually paid all at once, but it can also be paid in installments throughout the year.
The amount of the premium is determined by factors such as the type of insurance, the level of coverage, and the policyholder's risk profile.
The face value of the policy is $75,000. Since the annual premium rate is $2.34 per $1000 of face value, the annual premium Jesse would have to pay is:
$75,000 / $1000 = 75
75 x $2.34 = $175.50
Therefore, Jesse’s annual premium would be $175.50. The correct option is a.
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