Find the equation of a sphere if one of its diameters has endpoints: (-8, -6, -17) and (12, 14, 3).

stratsticks57jl
2022-07-29
Answered

Find the equation of a sphere if one of its diameters has endpoints: (-8, -6, -17) and (12, 14, 3).

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umgangistbf

Answered 2022-07-30
Author has **12** answers

The center of the sphere is the midpoint between the given points.

$(\frac{(-8)+(12)}{2},\frac{(-6)+(14)}{2},\frac{(-17)+(3)}{2})=(2,4,-7)$

the radius is the distance from the center to either point.

$\sqrt{(12-2{)}^{2}+(14-4{)}^{2}+(3+7{)}^{2}}=\sqrt{300}$

since the equation of a sphere with center (H,K,L) and radius r is.

$(x-H{)}^{2}+(y-K{)}^{2}+(z-L{)}^{2}={r}^{2}$

then the equation is.

$(x-2{)}^{2}+(y-4{)}^{2}+(z+7{)}^{2}+300$

$(\frac{(-8)+(12)}{2},\frac{(-6)+(14)}{2},\frac{(-17)+(3)}{2})=(2,4,-7)$

the radius is the distance from the center to either point.

$\sqrt{(12-2{)}^{2}+(14-4{)}^{2}+(3+7{)}^{2}}=\sqrt{300}$

since the equation of a sphere with center (H,K,L) and radius r is.

$(x-H{)}^{2}+(y-K{)}^{2}+(z-L{)}^{2}={r}^{2}$

then the equation is.

$(x-2{)}^{2}+(y-4{)}^{2}+(z+7{)}^{2}+300$

Alonzo Odom

Answered 2022-07-31
Author has **4** answers

The diameter length is

$2R=\sqrt{(-8-12{)}^{2}+(-6-14{)}^{2}+(-17-3{)}^{2}}=20\sqrt{3}$

$R=10\sqrt{3}$

the sphere equation is:

the center of the two points also the center of the sphere:

${x}_{c}=\frac{-8+12}{2}=2,{y}_{c}=\frac{-6+14}{2}=4,{z}_{c}=\frac{-17+3}{2}=-7$

so the equation of the sphere is:

$(x-2{)}^{2}+(y-4{)}^{2}+(z+7{)}^{2}={R}^{2}=300$

$2R=\sqrt{(-8-12{)}^{2}+(-6-14{)}^{2}+(-17-3{)}^{2}}=20\sqrt{3}$

$R=10\sqrt{3}$

the sphere equation is:

the center of the two points also the center of the sphere:

${x}_{c}=\frac{-8+12}{2}=2,{y}_{c}=\frac{-6+14}{2}=4,{z}_{c}=\frac{-17+3}{2}=-7$

so the equation of the sphere is:

$(x-2{)}^{2}+(y-4{)}^{2}+(z+7{)}^{2}={R}^{2}=300$

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My attempt : used base division method to find the area of ASN but I get extra variables which is tough...

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I'm defining my rhombus as follows: $[(0,0),(a,0),(b,c),(a+b,c)]$

I've managed to figure out that $c=\sqrt{{a}^{2}-{b}^{2}}$ and that the slopes of the diagonals are $\frac{\sqrt{{a}^{2}-{b}^{2}}}{a+b}$ and $\frac{-\sqrt{{a}^{2}-{b}^{2}}}{a-b}$

What I can't figure out is how they can be negative reciprocals of one another.

I mean to say that I could not find the algebraic proof. I've seen and understand the geometric proof, but I needed help translating it into coordinate form.

I'm defining my rhombus as follows: $[(0,0),(a,0),(b,c),(a+b,c)]$

I've managed to figure out that $c=\sqrt{{a}^{2}-{b}^{2}}$ and that the slopes of the diagonals are $\frac{\sqrt{{a}^{2}-{b}^{2}}}{a+b}$ and $\frac{-\sqrt{{a}^{2}-{b}^{2}}}{a-b}$

What I can't figure out is how they can be negative reciprocals of one another.

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Here are the five postulates:

1. Each pair of points can be joined by one and only one straight line segment.

2. Any straight line segment can be indefinitely extended in either direction.

3. There is exactly one circle of any given radius with any given center.

4. All right angles are congruent to one another.

5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which the angles are less than two right angles.

Questions:

1. These to me sounds more like something that shouldn't require proving... does it?

2. Why is it important to stress things that are obvious? For example, what other answers can you get when extending a line segment other than it can be extended indefinitely in either direction?

3. Similarly, what space can allow two circle of the same radius and center to be not the same?

4. Saying all right angles are congruent ... isn't that the same as saying all 64.506 degree angles are congruent? Isn't it ANY angle are congruent if they are the same degrees measures from the same reference point (say x-axis)?

5. Why do we need the 5th postulate?

1. Each pair of points can be joined by one and only one straight line segment.

2. Any straight line segment can be indefinitely extended in either direction.

3. There is exactly one circle of any given radius with any given center.

4. All right angles are congruent to one another.

5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which the angles are less than two right angles.

Questions:

1. These to me sounds more like something that shouldn't require proving... does it?

2. Why is it important to stress things that are obvious? For example, what other answers can you get when extending a line segment other than it can be extended indefinitely in either direction?

3. Similarly, what space can allow two circle of the same radius and center to be not the same?

4. Saying all right angles are congruent ... isn't that the same as saying all 64.506 degree angles are congruent? Isn't it ANY angle are congruent if they are the same degrees measures from the same reference point (say x-axis)?

5. Why do we need the 5th postulate?

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is rejected 'a priori' in the sense that we can only prove the validity of $A$ or $\mathrm{\neg}A$. My question is: given the rejection of these principles, in particular the second, is it still possible to keep the validity of the mathematical analysis within the intuitionism? Or maybe, accepting the intuitionistic approach, you need to develop a new kind of analysis, because, for example, it's impossible to use the 'reductio ad absurdum' to prove a theorem? Thanks

$A\vee \mathrm{\neg}A$

is rejected 'a priori' in the sense that we can only prove the validity of $A$ or $\mathrm{\neg}A$. My question is: given the rejection of these principles, in particular the second, is it still possible to keep the validity of the mathematical analysis within the intuitionism? Or maybe, accepting the intuitionistic approach, you need to develop a new kind of analysis, because, for example, it's impossible to use the 'reductio ad absurdum' to prove a theorem? Thanks