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3 years ago

The figure below shows concentric circles, both centered at 0.0

Mathematics

1 answer:

Rama09 [41]3 years ago

7 0

Answer:

C. 18 cm

Step-by-step explanation:

The ratio of the sides of the triangle shown is 12 : 15 = 4 : 5. We know it is a right triangle, so we know the missing side length completes the ratio

3 : 4 : 5 = 9 : 12 : 15

Half of XY is 9 cm, so the length of the entire chord is 18 cm.

_____

The chord is tangent to the inner circle, so makes a 90° angle with the radius to that tangent point. This tells you that the triangle shown is a right triangle. It also tells you that the short radius bisects the chord. The Pythagorean theorem can be used to find the length of the side not shown (half the chord length).

The unknown side (a) can be found from ...

15² = 12² +a²

225 -144 = a² . . . . . . subtract 12²

81 = a² . . . . . . . . . . . simplify

9 = a . . . . . . . . . . . . . take the square root

The chord length is 2a, so is ...

2(9 cm) = 18 cm . . . . length of chord XY

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Step-by-step explanation:

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Find the distance from the point (1,4) to the line y = 1/3x - 3

Troyanec [42]

Answer:

Step-by-step explanation:

If I'm not mistaken, and I very well could be, this is a calculus problem(?). In order to find the distance without calculus you'd need a point on the given line to use to find the distance in the distance formula. But you don't have a point on the given line, so we can find the shortest distance between the point (1, 4) and the given line using the derivative of the polynomial formed when using the distance formula.

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ and we have the x and y for x2 (or x1...it doesn't matter which you choose to fill in):

$d=\sqrt{(1-x)^2+(4-y)^2}$

but what we find is that we have too many unknowns here, namely, the distance, the x coordinate, and the y coordinate. So we can replace the y coordinate with what y is equal to in terms of the linear equation:

$d=\sqrt{(1-x)^2+(4-\frac{1}{3}x-3)^2 }$ and simplify:

$d=\sqrt{(1-x)^2+(7-\frac{1}{3}x)^2 }$

. No we'll expand each binomial by squaring:

$d=\sqrt{(1-2x+x^2)+(49-\frac{14}{3}x+\frac{1}{9}x^2) }$

. Combining like terms gives us

$d=\sqrt{\frac{10}{9}x^2-\frac{20}{3}x+50 }$

The distance between the point (1, 4) and the given line will be at a minimum when the polynomial above is at a minimum. We find the value of x for which the polynomial is at a minimum by finding its derivative, setting the derivative equal to 0, and then solving for x. The derivative of the polynomial is

$\frac{20}{9}x-\frac{20}{3}$

Setting equal to 0 and getting rid of the denominators gives us

20x - 60 = 0

Solving for x gives us

20x = 60 and x = 3.

That's the value of x that gives us the shortest distance between (1, 4) and the line y = 1/3x - 3. Sub into the distance formula that x value to find the distance:

$d=\sqrt{(\frac{10}{9})(3)^2-(\frac{20}{3})(3)+50 }$

which simplifies down, finally, to

x ≈ 6.325 units

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3 years ago