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We can use the property that the product of the lengths of the altitudes of a triangle is equal to the product of its semiperimeter and its inradius. In other words:

AD * BE * CF = s * r

where s is the semiperimeter of the triangle and r is its inradius. Since CF is the largest of the three altitudes, we want to find the largest possible value of CF, which means we want to maximize the right-hand side of this equation.

Let's label the sides of triangle ABC as a, b, and c, with opposite vertices A, B, and C, respectively. Then, the semiperimeter s is:

s = (a + b + c) / 2

The inradius can be found using the formula:

r = A / s

where A is the area of the triangle. We can find the area of the triangle using any of the altitudes, so let's use AD:

A = (1/2) * AD * BC

Substituting the given values, we get:

A = (1/2) * 12 * BC = 6BC

Now we can substitute the expressions for s and r into the equation above:

AD * BE * CF = s * r
12 * BE * CF = [(a + b + c) / 2] * (6BC / (a + b + c))

Simplifying, we get:

24BE * CF= 3BC * 6BE

Dividing both sides by 6BE, we get:

4CF = BC

To maximize CF, we want BC to be as large as possible. However, we also know that BC must be less than the sum of the other two sides, since it is opposite the largest angle of the triangle. Therefore, we want BC to be as close as possible to the sum of the other two sides.

Let's assume that the sum of the other two sides is equal to 2x, where x is a positive integer. Then, BC must be less than 2x. We want BC to be as close as possible to 2x, so we choose BC = 2x - 1. This ensures that BC is as large as possible while still being less than 2x.

Now we can substitute this value of BC into the equation we derived earlier:

4CF = BC = 2x - 1

Since x is a positive integer, the largest possible value of BC is when x = 5, which gives BC = 9. Therefore, the largest possible value of CF is:

4CF = 2x - 1 = 9 * 4 = 36

So the largest possible value of CF is 36. 

If the slope on graph paper has a slope of 4/9, this means that for every 4 units in the vertical direction, there are 9 units in the horizontal direction. We can use this ratio to find the length of the slope in meters.

Let's assume that the slope rises 4 units in the vertical direction for every 9 units in the horizontal direction. This means that the slope has a rise of 4 cubes and a run of 9 cubes on the graph paper. Since each cube represents 100 meters, the rise of the slope is 4 x 100 = 400 meters and the run of the slope is 9 x 100 = 900 meters.

To find the length of the slope, we can use the Pythagorean theorem:

length^2 = rise^2 + run^2

Substituting the values we found, we get:

length^2 = (400)^2 + (900)^2

Simplifying, we get:

length^2 = 160000 + 810000

length^2 = 970000

Taking the square root of both sides, we get:

length = sqrt(970000) = 985 meters (rounded to the nearest meter)

Therefore, the length of the slope is approximately 985 meters.

We can use the property of similar triangles to solve this problem. The triangles formed by the poles and their shadows are similar, so the ratios of corresponding sides are equal. We can use this property to find which of the given options are plausible.

Let h1 and h2 be the heights of the tether ball pole and the flagpole, respectively, and let x1 and x2 be the lengths of their shadows. Then, we have:

h1 / x1 = h2 / x2

We know that h1 = 3 m and h2 = 6.8 m, so we can substitute these values:

3 / x1 = 6.8 / x2

Multiplying both sides by x1 x2, we get:

3x2 = 6.8x1

Simplifying, we get:

x1 = (3/6.8) x2

This means that the length of the tether ball pole's shadow is 3/6.8 times the length of the flagpole's shadow.

Now, we can check which of the given options satisfy this condition:

A) tether 1.35m, flag 3.4m
The ratio of their lengths is 1.35/3.4 = 0.397. This does not satisfy the condition.

B) tether 1.8m, flag 4.08m
The ratio of their lengths is 1.8/4.08= 0.441. This does not satisfy the condition.

C) tether 3.75m, flag 8.35m
The ratio of their lengths is 3.75/8.35 = 0.449. This satisfies the condition.

D) tether 0.6m, flag 1.36m
The ratio of their lengths is 0.6/1.36 = 0.441. This does not satisfy the condition.

E) tether 2m, flag 4.8m
The ratio of their lengths is 2/4.8 = 0.417. This does not satisfy the condition.

Therefore, the lengths of the shadows that could be plausible are C) tether 3.75m, flag 8.35m. 

To solve this system, we can use algebraic manipulation to eliminate variables and solve for one variable in terms of the others.

First, let's simplify the second equation by moving the q term to the left-hand side:

q - 2r - q = -2

Simplifying, we get:

-q - 2r = -2

Now we can eliminate q by adding the first equation to this simplified second equation:

(p - 2q) + (-q - 2r) = 3 - 2

Simplifying, we get:

p - 3q - 2r = 1

Next, we can use the third equation to eliminate r by solving for p in terms of r and substituting into the third equation:

p + r = 9 + p

Subtracting p from both sides, we get:

r = 9

Now we can substitute this value of r into the equation we just derived:

p - 3q - 2r = 1

p - 3q - 2(9) = 1

Simplifying, we get:

p - 3q = 19

Finally, we can use the first equation to solve for p in terms of q:

p - 2q = 3

p = 2q + 3

Substituting this expression for p into the equation we just derived, we get:

(2q + 3) - 3q = 19

Simplifying, we get:

-q = 16

Dividing both sides by -1, we get:

q = -16

Now we can use the expression we derived earlier to solve for p:

p = 2q + 3 = 2(-16) + 3 = -29

Therefore, the ordered triple that satisfies the system is (-29, -16, 9).

We can solve this problem by using the second equation to solve for a in terms of b, and then substituting that expression for a into the first equation.

From the equation ab = 72, we can solve for a by dividing both sides by b:

a = 72/b

Now we can substitute this expression for a into the first equation:

ab^4 = 48

(72/b) * b^4 = 48

Simplifying, we get:

72b^3 = 48

Dividing both sides by 72, we get:

b^3 = 2/3

Taking the cube root of both sides, we get:

b = (2/3)^(1/3)

This is the exact value of b in the ordered pair (a, b). If you need a numerical approximation, you can use a calculator to evaluate the cube root of 2/3, which is approximately 0.892. Therefore, b is approximately 0.892 in this ordered pair. 

1. To find the area of triangle ABC, we can use the formula:

Area = (1/2) * base * height

In this case, we can take AB as the base and BC as the height, since they are perpendicular. Therefore:

Area = (1/2) * AB * BC = (1/2) * 6 * 8 = 24 square units.

So the area of triangle ABC is 24 square units.

2. To find angle BAC, we can use the fact that the angles in a triangle add up to 180 degrees. Therefore:

angle BAC + angle ABC + angle ACB = 180 degrees

Substituting the given values, we get:

angle BAC + 135 degrees + 30 degrees = 180 degrees

Simplifying, we get:

angle BAC = 180 degrees - 135 degrees - 30 degrees = 15 degrees

So angle BAC is 15 degrees.

We can prove this using the Pythagorean Theorem and some algebraic manipulation.

First, let's label the sides of the right triangle as follows:

- a is the length of the side opposite angle A
- b is the length of the side opposite angle B
- c is the length of the hypotenuse
- h is the length of the altitude from C to AB
- x is the length of AH
- y is the length of BH

Since ABC is a right triangle, we know that a^2 + b^2 = c^2. We can also use the fact that the altitude from C to AB divides the triangle into two similar right triangles, so we have:

x/h = h/y

Solving for x and y, we get:

x = h^2 / y

y = h^2 / x

Substituting these expressions for x and y into the equation we want to prove, we get:

[(h^2 / y) + h]^2 * [(h^2 / x) + h]^2 = (a + b)^4

Simplifying the left-hand side, we can factor out h^4 and get:

h^4 * [(1/y + 1/h)^2] * [(1/x + 1/h)^2] = (a + b)^4

Multiplying both sides by [(xy)/h^4]^2, we get:

(x + h)^2 *(y + h)^2 = (a + b)^4

This completes the proof.

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I'm doing inequality, i need to graph it

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