Question

The triangle has a point of concurrency at P.

A triangle has a point of concurrency at P. Lines are drawn from the points of the triangle to point P. Lines are drawn from point P to the sides of the triangle to form right angles. The length of the line segment to the left side of the triangle is 24. The length of the line segment to the bottom side of the triangle is 3 x + 3. The length of the line segment from point P to the left corner of the triangle is 5 x minus 4.
Find the value of x that would make P the incenter of the triangle.

x =



Find the value of x that would make P the circumcenter of the triangle.

x =

Answer 1

Incenter: The incenter is equidistant from the sides, so the perpendicular distances to the left and bottom sides must be equal:
24 = 3x + 3 -> 3x = 21 -> x = 7.

Circumcenter: The circumcenter is equidistant from the vertices, so the given vertex distances must be equal. Equating the two vertex distances 24 and 5x − 4 gives
24 = 5x − 4 -> 5x = 28 -> x = 28/5.

Answers:
- For the incenter: x = 7.
- For the circumcenter: x = 28/5.