Finding unknown value: rt triangle w/ pts (x, x + 1), (x + 2, x + 3), (x + 3, 2x + 4)

Danaher

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Hi everyone,

I'm trying to teach myself math when I'm not at work and don't have access to a tutor. I'm currently working on Geo/Trig.

In my textbook, questions 1-8 are straight forward and simple variations of each other, finding the distance between coordinates on a graph. The last question of the list is very different.

The question: "Find x so that points A (x, x + 1), B (x + 2, x + 3) and C (x + 3, 2x + 4) form a right-angled triangle.

The answer in the textbook is: "x = -2 or -6"

Knowing the answer hasn't been able to help me determine where to even start on this.

An initial thought of where to begin was to determine which of the two points form the hypotenuse - which I think would be AC. Even if AC is the hypotenuse, I'm unsure how to proceed.

If someone would like to explain the process of understanding this, it would be much appreciated. Thank you.
 
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Hi everyone,

I'm trying to teach myself math when I'm not at work and don't have access to a tutor. I'm currently working on Geo/Trig.

In my textbook, questions 1-8 are straight forward and simple variations of each other, finding the distance between coordinates on a graph. The last question of the list is very different.

The question: "Find x so that points A:(x, x + 1), B:(x + 2, x + 3) and C:(x + 3, 2x + 4) form a right-angled triangle.

The answer in the textbook is: "x = -2 or -6"

Knowing the answer hasn't been able to help me determine where to even start on this.

An initial thought of where to begin was to determine which of the two points form the hypotenuse - which I think would be AC. Even if AC is the hypotenuse, I'm unsure how to proceed.

If someone would like to explain the process of understanding this, it would be much appreciated. Thank you.
Why do you need to know the hypotenuse up front? Can you determine the SLOPE of line AB and line AC an line BC?
 
Hi everyone,

I'm trying to teach myself math when I'm not at work and don't have access to a tutor. I'm currently working on Geo/Trig.

In my textbook, questions 1-8 are straight forward and simple variations of each other, finding the distance between coordinates on a graph. The last question of the list is very different.

The question: "Find x so that points A: (x, x + 1), B: (x + 2, x + 3) and C: (x + 3, 2x + 4) form a right-angled triangle.

The answer in the textbook is: "x = -2 or -6"

Knowing the answer hasn't been able to help me determine where to even start on this.

An initial thought of where to begin was to determine which of the two points form the hypotenuse - which I think would be AC. Even if AC is the hypotenuse, I'm unsure how to proceed.

If someone would like to explain the process of understanding this, it would be much appreciated. Thank you.

Whether you use slopes (for perpendicularity, which will be much easier) or lengths (with the Pythagorean theorem, which is what the context evidently suggests), the key idea will be to try each possibility for the hypotenuse and see if you can solve for x. It will turn out that two choices yield solutions, and the third does not.

Key idea: when you don't know something about a problem, pick a possibility and continue, then come back and try other cases if necessary.
 
Thank you both for responding.

I should have mentioned I'm supposed to solve this using Pythagorean theorem. I'm on chapter 1 of the book and the only two forumla given so far are "A^2 + B^2 = C^2" and the equation for calculating distance "(x-x)^2 + (y-y)^2"

Questions 1-8 are all "Find the distance between each of the following pairs of points: eg. (8,5) and (2,-3)".


@Dr.Peterson,

"the key idea will be to try each possibility for the hypotenuse and see if you can solve for x"

I'm not going to lie, I don't completely understand what that means. Surely I don't just plug in a random value for x?
 
I've been staring at this problem totaling over an hour, I have no idea how to begin. I must have some fundamental misunderstanding of this.
 
I should have mentioned I'm supposed to solve this using Pythagorean theorem....

I've been staring at this problem totaling over an hour, I have no idea how to begin. I must have some fundamental misunderstanding of this.
What have you gotten for the lengths of the possible sides, being AB, BC, and AC?

When you picked pairs of lengths to be the legs, set up the equation from the Pythagorean Theorem, and tried to solve the resulting equation, what did you get? (Two of the pairings work; one does not.)

Please be complete. Thank you! ;)
 
Thank you both for responding.

I should have mentioned I'm supposed to solve this using Pythagorean theorem. I'm on chapter 1 of the book and the only two forumla given so far are "A^2 + B^2 = C^2" and the equation for calculating distance "(x-x)^2 + (y-y)^2"

Questions 1-8 are all "Find the distance between each of the following pairs of points: eg. (8,5) and (2,-3)".


@Dr.Peterson,

"the key idea will be to try each possibility for the hypotenuse and see if you can solve for x"

I'm not going to lie, I don't completely understand what that means. Surely I don't just plug in a random value for x?

I don't know why I didn't see this reply three days ago. Sorry.

You have A (x, x + 1), B (x + 2, x + 3) and C (x + 3, 2x + 4).

First write an expression for each distance: AB, BC, AC, using the distance formula you were taught.

Since you don't know which side will be the hypotenuse, you have to try each possibility for which it is -- whether the hypotenuse is AB, BC, or AC.

If AB is the hypotenuse, then you have to have AB^2 = BC^2 + AC^2. Replace each length with the expression you found, and solve the equation. That will give you one answer (if you can solve it). Then repeat, assuming the hypotenuse is BC or AC.
 
I've been staring at this problem totaling over an hour, I have no idea how to begin. I must have some fundamental misunderstanding of this.


first draw a picture of a triangle with pts at the corners so you can visualize the problem.
find the distance between each pt using pathagorean theorem.
e.g. (xA-xB)2 + (yA-yB)2=distanceAB2
then use distances in path. theorem.
 
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I don't know why I didn't see this reply three days ago. Sorry.

You have A (x, x + 1), B (x + 2, x + 3) and C (x + 3, 2x + 4).

First write an expression for each distance: AB, BC, AC, using the distance formula you were taught.

Since you don't know which side will be the hypotenuse, you have to try each possibility for which it is -- whether the hypotenuse is AB, BC, or AC.

If AB is the hypotenuse, then you have to have AB^2 = BC^2 + AC^2. Replace each length with the expression you found, and solve the equation. That will give you one answer (if you can solve it). Then repeat, assuming the hypotenuse is BC or AC.

My comment had to be approved by a moderator. The view-able date was later than the post date.

If I'm understanding correctly, that would make 'distance AB' = √(x+2-x)^2 + (x+3-x-1)^2

Which simplifies to = √(2)^2 + (2)^2

Giving 'distance AB' = √8

- With the presupposition I'm on the right track above, I'll continue with AC -

'distance AC' = √(x+3-x)^2 + (2x+4-x+1)^2

=
√(3)^2 + (x+5)^2

=
√9 + (x+5)^2

Thank you very much for your time.
 
My comment had to be approved by a moderator. The view-able date was later than the post date.

Yes, but I should have known to look ...

If I'm understanding correctly, that would make 'distance AB' = √(x+2-x)^2 + (x+3-x-1)^2

Which simplifies to = √(2)^2 + (2)^2

Giving 'distance AB' = √8

- With the presupposition I'm on the right track above, I'll continue with AC -

'distance AC' = √(x+3-x)^2 + (2x+4-x+1)^2

=
√(3)^2 + (x+5)^2

=
√9 + (x+5)^2

Thank you very much for your time.

You just made one sign error here; AC should be √[(x+3-x)^2 + (2x+4-x-1)^2] = √[9 + (x+3)^2] = √[x^2+6x+18].

Now do the same for BC, and use the three expressions to write each of the three equations,

AB^2 + BC^2 = AC^2
BC^2 + AC^2 = AB^2
AC^2 + AB^2 = BC^2

and try solving each.

The method using slopes is far easier, but this is a good challenge!
 
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