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

Answer: x= 7/2 would make P the circumcenter of the triangle.

Step-by-step explanation: To find the value of x that would make point P the circumcenter of the triangle, we need to use the property that the circumcenter is the point of concurrency for the perpendicular bisectors of the sides of the triangle.

Since the triangle's perpendicular bisectors are drawn from point P to the sides of the triangle to form right angles, the length from P to the midpoint of a side is equal to the circumradius (R), which is the distance from the circumcenter to any vertex of the triangle.

We are given that the length of the line segment to the left side of the triangle is 24, which is the circumradius (R).

The length of the line segment to the bottom side of the triangle is 3x+3.

The length of the line segment from point P to the left corner of the triangle is 5x−4.

For the circumcenter of the triangle, the length from the circumcenter to a side is equal to the same length from the circumcenter to the other two sides.

So, we have the following equation:

R=(3x+3)=(5x−4)

Now, let's solve for x:

3x + 3 = 5x − 4

Subtract 3x from both sides:

3=2x−4

Now, add 4 to both sides:

7=2x

Finally, divide both sides by 2 to find the value of x:

x=7/2

Answer 2

incenter= 7

circumcenter= 6

Explanation: