In \triangle ABC, the side lengths are AB = 12, BC = 15, and AC = 18. A point P lies stric… — ACT

In $\triangle ABC$ , the side lengths are $AB = 12$ , $BC = 15$ , and $AC = 18$ . A point $P$ lies strictly inside $\triangle ABC$ . If the lengths of the segments $BP$ and $CP$ are both integers, what is the maximum possible value of the sum of these two lengths?

A
26
B
27
28Answer
D
29
E
30

Answer

The correct answer is 28. According to the properties of triangles, for any point

P

strictly inside

\triangle ABC

, the sum of the interior segments is strictly less than the sum of the other two sides:

BP + CP < AB + AC = 12 + 18 = 30

. Since

BP

and

CP

are integers, we check the boundary where

P

lies on the side

AC

at an integer distance

CP = z

from

C

. Applying Stewart's Theorem, the boundary length

BP = y_{bound}

\sqrt{z^2 - 22.5z + 225}

. For the largest possible integer value

z = 17

, the boundary value is approximately

11.47

. Because the point must lie strictly inside the triangle,

BP

must be strictly less than this boundary, so the maximum integer value for

BP

is 11. This yields a maximum sum of

11 + 17 = 28

. Lower integer values of

z

yield smaller maximum sums (for example, if

z = 16

, the boundary is exactly 11, so

BP

can be at most 10, giving a sum of 26).

Step-by-Step Solution

Apply the interior point triangle inequality theorem.

For any point

P

strictly inside

\triangle ABC

, the sum of the distances to two vertices is strictly less than the sum of the other two sides:

BP + CP < AB + AC

This establishes the theoretical upper bound for the sum of the two segment lengths.

Calculate the theoretical upper bound.

Since

AB = 12

and

AC = 18

, we have

BP + CP < 12 + 18 = 30

. Since

BP

and

CP

must be integers, the sum

BP + CP

can be at most 29.

This sets the initial integer limit before evaluating if it is geometrically possible.

Analyze the boundary conditions for integer lengths using Stewart's Theorem.

Let

CP = z

and

BP = y

, where

y

and

z

are integers. As

P

approaches the side

AC

, the boundary value

y_{bound}

represents the distance from

B

to a point on

AC

at distance

z

from

C

. Using Stewart's Theorem, this boundary satisfies:

y_{bound}^2 = z^2 - 22.5z + 225

. Since

P

is strictly inside the triangle,

y

must be strictly less than

y_{bound}

This provides the mathematical relationship determining whether a point is inside the triangle for any given integer length of one segment.

Test the maximum possible integer value for

z

to maximize the sum

y + z

Since

P

is strictly inside,

z

must be strictly less than

AC = 18

, so the maximum integer for

z

is 17. For

z = 17

, the boundary value is

y_{bound} = \sqrt{17^2 - 22.5(17) + 225} = \sqrt{131.5} \approx 11.47

. Since

y < y_{bound}

, the maximum integer value for

y

is 11. This yields a maximum sum of

11 + 17 = 28

This determines the actual maximum integer sum that can be geometrically realized within the triangle.

Key Concept

Triangle Inequality Theorem and Interior Point Properties

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