Find the Max of $PC$ in $\triangle ABC$

MHB
Thread starter Albert1
Start date Jul 10, 2014
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Max pc

In summary, the maximum value of $PC$ in $\triangle ABC$ is equal to the length of the longest side, or the hypotenuse. This can be found using the Pythagorean theorem, and it cannot be greater than the sum of the other two sides. The position of point $C$ greatly affects the maximum value of $PC$, with it being at its highest when located on the opposite side of the triangle from point $A$.

Jul 10, 2014

Albert1

$\triangle ABC$ with $AB=BC=CA$
if another point $P$ and $PA=2, \,\, PB=3$
please find :$max(PC)$

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Jul 14, 2014

Albert1

Albert said:

$\triangle ABC$ with $AB=BC=CA$ if another point $P$ and $PA=2, \,\, PB=3$ please find :$max(PC)$

my solution:

Related to Find the Max of $PC$ in $\triangle ABC$

1. What is the maximum value of $PC$ in $\triangle ABC$?

The maximum value of $PC$ occurs when the point $C$ is located on the opposite side of the triangle from the point $A$. In this case, $PC$ is equal to the length of the longest side of the triangle, also known as the hypotenuse.

2. How do you find the maximum value of $PC$ in $\triangle ABC$ using the Pythagorean theorem?

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. In $\triangle ABC$, $PC$ is the hypotenuse, so we can use the Pythagorean theorem to find its maximum value by squaring the lengths of sides $AB$ and $AC$, then taking the square root of the sum.

3. Can the maximum value of $PC$ be greater than the sum of the lengths of sides $AB$ and $AC$?

No, the maximum value of $PC$ cannot be greater than the sum of the lengths of sides $AB$ and $AC$. This is because in a triangle, any side must be shorter than the sum of the other two sides. Therefore, the hypotenuse $PC$ cannot be longer than the sum of $AB$ and $AC$.

4. What is the relationship between the maximum value of $PC$ and the angles of $\triangle ABC$?

The maximum value of $PC$ is directly related to the angles of $\triangle ABC$. In a right triangle, the length of the hypotenuse is always greater than the lengths of the other two sides, and this relationship holds true for any type of triangle. As the angles of $\triangle ABC$ increase, the maximum value of $PC$ also increases, and vice versa.

5. How does the position of point $C$ affect the maximum value of $PC$ in $\triangle ABC$?

The position of point $C$ greatly affects the maximum value of $PC$ in $\triangle ABC$. As mentioned earlier, when $C$ is located on the opposite side of the triangle from $A$, $PC$ is at its maximum value. However, if $C$ is located on the same side as $A$, then $PC$ will be shorter. In general, the closer $C$ is to $A$, the shorter $PC$ will be, and the farther away $C$ is, the longer $PC$ will be.