MATHEMATICS HIGH SCHOOL

The three sides of a triangle are tangent to a unique circle called the incircle. On the coordinate plane, the incircle of ABC has its center at the origin. The lines whose equations are x+4y=6s(qrt17), 2x+(sqrt5)y=-18, and 2(sqrt2)x=y+18 contain the sides of ABC. What is the length of the radius of the incircle?sqrt= square root

A. 2 units
B. 3 units
C. 6 units
D. 8 units

Answers

Answer 1

Answer:

We are looking for the length of a radius.

If we can find the coordinates of the two endpoints of a radius, then we can find its length.

A tangent to a circle intersects the circle at a single point.

A radius of the circle drawn to the point of tangency is perpendicular to the tangent.

We know the center of the circle is the origin. That means we know one endpoint of every radius.

We use one side of the triangle which is given as an equation.

We find its slope. Then using that slope and the coordinates of the origin,

we can find the equation of the radius that is perpendicular to that side.

We then solve the equations simultaneously to find the coordinates of the point of tangency. The point of tangency is the other endpopint of the radius.

Knowing two endpoints of a radius, we can find its length. That will give us the answer since we are looking for the length of the radius.

Let's use the first given equation.

Find its slope.

$x + 4y = 6 √(17)$

$4y = -x + 6 √(17)$

$y = -(1)/(4)x + (3√(17))/(2)$

The slope of the side of the triangle is -1/4.

The slope of the radius perpendicular to that side is 4.

The equation of the line that contains the radius is

$y - y_1 = m(x - x_1)$

We are told the circle is centered at the origin, so one point on the line containing the radius is (0, 0).

$y - 0 = 4(x - 0)$

$y = 4x$

The line containing the radius is y = 4x.

Now we use the equation of the line containing the side of the triangle and the equation of the line containing the radius as a system of equations to find their point of intersection. That point is the other endpoint of the radius.

$x + 4y = 6 √(17)$

$y = 4x$

Multiply the first equation by 4 and subtract the x-term from both sides.

$16y = -4x + 24 √(17)$

$y = 4x$

Add the equations and solve for y.

$17y = 24 √(17)$

$y = (24 √(17))/(17) = (24)/(√(17))$

Now replace y with the value we found, and solve for x in the second equation.

$(24 √(17))/(17) = 4x$

$(6 √(17))/(17) = x$

$x = (6 √(17))/(17) = (6)/(√(17))$

The coordinates of the point of intersection are

$((6)/(√(17)), (24)/(√(17)))$

This point of intersection is one endpoint of the radius.

The other endpoint of the radius is the origin.

Now we need to find the the length of the radius.

$r = √((x_2 - x_1)^2 + (y_2 - y_1)^2)$

$r = \sqrt{((6)/(√(17)) - 0)^2 + ((24)/(√(17)) - 0)^2}$

$r = \sqrt{(36)/(17) + (576)/(17)}$

$r = \sqrt{(612)/(17)}$

$r = √(36)$

$r = 6$

Answer: The radius is 6 units long, choice C.

Answer 2

Answer:

Answer:

Step-by-step explanation:

edge21

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Answers

The quotient of a number and it's reciprocal is square of the original number

Given :

The quotient of a number and it's a reciprocal

For example we take a number 3

Let the number be 3

The reciprocal of 3 is $(1)/(3)$

The quotient of a number 3 and it's a reciprocal $(1)/(3)$ is $(3)/((1)/(3) ) =3 \cdot (3)/(1) =9$

we know that $3^2=9$

Square of 3 is 9

From this we can say that,

The quotient of a number and it's a reciprocal is the square of the original number.

Learn more : brainly.com/question/1928559

The quotient of a number and its reciprocal is the square of the original number.

The sum of a number and 5, less than 10, is 3.

Answers

The sum of a number and 5, less than 10, is 3 can be expressed as:

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where x is unknown number.

x + 5 = 3 + 10
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What's bigger 25,000
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Answers

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if you could show yourworkthat would be great :3

Answers

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which in turn means

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Laura rents a movie for a flat fee of $2.00 plus an additional $0.50 for each night she keeps the movie. Choose the cost function that represents this scenario if x equals the number of nights Laura has the movie.a) c(x) = 2.00x + 0.50
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Answers

b)c(x) = 2.00 + 0.50x

Remember that since 2.00 is the flat fee, so there is no variable. After that, though, there is the fee of .50 which is charged every day you have a movie.

say you had the movie rented out for five days
c(5)=2.00+.50(5)
=2.00+2.5
c(5)=4.5

Hope this helped!

Answer:

The cost function that represents this scenario is c(x) = 2 + 0.50x .

Option (b) is correct .

Step-by-step explanation:

As given

Laura rents a movie for a flat fee of $2.00 plus an additional $0.50 for each night she keeps the movie.

if x equals the number of nights Laura has the movie.

Than the cost function that represents this scenario .

c(x) = Flat fee + Cost for x equals the number of nights Laura has the movie.

c(x) = 2 + x × 0.50

c(x) = 2 + 0.50x

Therefore the cost function that represents this scenario is c (x) = 2 + 0.50x .

Option (b) is correct .

Danny has a 15-foot board that he wants to use as a motorcycle ramp. If Danny wants the end of the ramp to be 4 feet off the ground, what angle will be formed between the board and the ground? Round the answer to the nearest whole degree.A. 14 degrees
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Answers

This forms a right triangle with hypotenuse of length 15 and vertical component of length 4. Use the inverse of sine (vertical/hypotenuse) to get arcsin(4/15) = 15.4 degrees which rounds to 15
Final answer:
Answer choice B, 15 degrees

Answer:15

Step-by-step explanation:

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