Altitude Problem

[neno89]

I need help solving this problem:

In triangle ABC, point A is at (5,4) and point B is at (-3,4). The triangle's altitude from point C to segment AB is 6 units long. Draw a sketch of point A and B, and state the equations of the two lines point C must lie on.

 

[stapel]

From just the given information, we can't answer the question absolutely, since C could be anywhere either along the line from (-3, -2) to (5, -2) or else along the line from (-3, 10) to (5, 10). Without further information (like "C is above the segment AB" or "ABC is isosceles", etc), I see no way to find "the" equations that C "must" lie on.

Sorry.

Eliz.

 

[Euler]

Draw your triangle. Now, point C and the altitude of the triangle are going to line up, due to the fact that an altitude runs right through one vertex of a triangle (which we know is C) and will be perpendicular to line segment AB.

What this is asking for, is what the equations of lines AC and BC will be.

Tell me, is the triangle any specific shape?

 

[stapel]

...the altitude...runs right through one vertex of a triangle (which we know is C) and will be parallel to line segment AB.

Shouldn't the altitude from C to AB be perpendicular to AB? Or have I mixed up definitions again...?

Eliz.

 

[Euler]

...the altitude...runs right through one vertex of a triangle (which we know is C) and will be parallel to line segment AB.

Shouldn't the altitude from C to AB be perpendicular to AB? Or have I mixed up definitions again...?

Eliz.

No you are correct, it was an error on my part, and I thank you for catching it.

 

[neno89]

no the triangle doesn't a specfic shape.

 

[stapel]

no the triangle doesn't a specfic shape.

Then I see no way to find what they're asking for. Sorry.

Eliz.

 

[Euler]

Yes, as stapel said, if the triangle does not have a specific shape, then the point C could be at any value -3>x>5 (and still fit the definition of altitude, all while being 6 units away from line AB. This gives you a wide range of values, but you will not get just two equations, you will get a lot.

 

[soroban]

Hello, neno891

I believe everyone missed the point of this question . . .

In triangle ABC, point A is at (5,4) and point B is at (-3,4).
The triangle's altitude from point C to segment AB is 6 units long.
Draw a sketch of point A and B,
and state the equations of the two lines point C must lie on.

                    C
        ......|.....*...... y = 10
              |     :
              |     :6
              |     :
              |     : 
    B - - - - + - - - - - - A
    *         |   :         *
 (-3,4)       |   :6      (5,4)
              |   :
      --------+--------------
        ......|...*........  y = -2
              |   C

If vertex C is to be 6 units from AB,
. . it can be anywhere on the two horizontal lines: y = 10 and y = -2.

And that is what the question asked for.

 

[stapel]

If vertex C is to be 6 units from AB, it can be anywhere on the two horizontal lines: y = 10 and y = -2.

"Anywhere"?

If C is at, say, (12, 10), then it is not six units from AB, because AB ended back at (5, 10), and C's "distance" from AB will no longer be the direct perpendicular measure.

Eliz.

 

[soroban]

Hello, Eliz!

The problem referred to an altitude of the triangle
. . which is measured, in this problem, from the vertex C to side AB (extended, if necessary).

[Vertex C can be 10 units or a light-year to the right or left.
. . The area of the triangle remains constant.]

 

[stapel]

The problem referred to an altitude of the triangle...from the vertex C to side AB (extended, if necessary).

But the exercise doesn't mention the possibility of extension; it specifically says "the segment AB".

Eliz.

 

[Gene]

But it will still be on y=10 or y=-2. The question doesn't ask where on those lines it is.
--------------------
Gene

 

[Euler]

But Gene, the equations will change because the slope will very depending on where C is in relation to points A and B. The closer C gets to B, the more rapid the slope gets for that particular equation. The further away, the more the slope decreases. You are not going to get only two equations.

 

[Gene]

I'll admit the problem is badly worded but the "only two lines point C must lie on" (or more properly the two lines on one of which point C must lie) are Sorobans y=10 or y=-2
Suppose point C is at (x,10). The slope is 6/(x-5) so the point-slope form is
y=(6/(x-5))*(x-5)+4
y = 10
QED :twisted:
------------------
Gene

 

[Denis]

I need help solving this problem:
In triangle ABC, point A is at (5,4) and point B is at (-3,4). The triangle's altitude from point C to segment AB is 6 units long. Draw a sketch of point A and B, and state the equations of the two lines point C must lie on.

I agree with soroban, and from the wording "equations of the 2 lines", it is
evident that this is meant:
"The triangle's altitude from point C is 6 units long."