Which of the following statements are true?
a. the incenter of a triangle is equidistant from all three vertices of that triangle
b. The incenter of a triangle always lies inside that triangle
c. the incenter of a triangle is the point of concurrency of the perpendicular bisectors of that triangle
d. the incenter of a triangle is the point of concurrency of the altitudes of that triangle.
A simple google on triangle incenter (or a quick review in your text) should make this easy to answer.
What did you find? (Hint: (c) is false)
Is it B? The incenter of a triangle always lies inside that triangle?
"The incenter is the last triangle center we will be investigating. It is the point forming the origin of a circle inscribed inside the triangle. Like the centroid, the incenter is always inside the triangle. It is constructed by taking the intersection of the angle bisectors of the three vertices of the triangle."
Thank you so much!! I got a 10/10!
Well, let's see if I can bring some humor to these statements:
a. The incenter of a triangle is equidistant from all three vertices of that triangle. Well, it's like being stuck in the middle of a three-way dance-off, trying to give everyone equal attention. So, true!
b. The incenter of a triangle always lies inside that triangle. It's like trying to find a comfy spot in a crowded subway train - it's always inside, trying not to step on anyone's toes. True again!
c. The incenter of a triangle is the point of concurrency of the perpendicular bisectors of that triangle. It's like a traffic jam of perpendicular lines, all meeting at one central point. So true!
d. The incenter of a triangle is the point of concurrency of the altitudes of that triangle. Imagine three stalkers following you around, trying to keep an eye on your every move. They all end up meeting at the same spot - creepy, but true!
So, to sum it up: a, b, c, and d are all true!
To determine which of the statements are true, let's go through each statement one by one and explain how to verify its correctness:
a. The statement that the incenter of a triangle is equidistant from all three vertices of that triangle is true. To check this, you can calculate the distances from the incenter to each vertex of the triangle. If all three distances are equal, then the statement is correct.
b. The statement that the incenter of a triangle always lies inside that triangle is also true. You can visually inspect the given triangle and see if the incenter is contained within its interior. Alternatively, you can prove it mathematically using properties of triangles and the definition of the incenter.
c. The statement that the incenter of a triangle is the point of concurrency of the perpendicular bisectors of that triangle is false. The point of concurrency of the perpendicular bisectors is the circumcenter, not the incenter. To verify this, you can construct the perpendicular bisectors of the sides of the triangle and see that they intersect at a different point from the incenter.
d. The statement that the incenter of a triangle is the point of concurrency of the altitudes of that triangle is also false. The point of concurrency of the altitudes is called the orthocenter, not the incenter. To confirm this, you can construct the altitudes of the triangle and observe that they intersect at a different location from the incenter.
In summary, the correct statements are:
a. The incenter of a triangle is equidistant from all three vertices of that triangle.
b. The incenter of a triangle always lies inside that triangle.
The incorrect statements are:
c. The incenter of a triangle is the point of concurrency of the perpendicular bisectors of that triangle.
d. The incenter of a triangle is the point of concurrency of the altitudes of that triangle.