Question

which type of triangle is the circumcenter located outside of?
a. right triangle
b. isosceles triangle
c. acute triangle
d. obtuse triangle

which point of concurrency is equidistant from all three vertices of a triangle?
a. orthocenter
b. circumcenter
c. incenter
d. centroid

what is the significance of the incenter in constructing an incircle?
a. the incenter is the midpoint of the medians
b. the incenter is the midpoint of the sides
c. the incenter is the center of the circumcircle
d. the incenter is the center of the incircle

which statement is true about the circumcircle of a triangle?
a. it passes through all the vertices of a triangle
b. its center is the centroid
c. it is tangent to all three sides
d. it is always inside the triangle

Explanation:

First Question:

To determine the triangle with the circumcenter outside, we analyze each type:

• A right triangle's circumcenter is on the hypotenuse (midpoint).
• An isosceles triangle's circumcenter location depends on its angles (can be inside or on the triangle if right - isosceles).
• An acute triangle's circumcenter is inside the triangle.
• An obtuse triangle has an angle greater than \(90^\circ\), and its circumcenter lies outside the triangle.

We recall the definitions of each point of concurrency:

• The orthocenter is the intersection of the altitudes.
• The circumcenter is the intersection of the perpendicular bisectors of the sides. By definition, the distance from the circumcenter to each vertex of the triangle is equal (it is the center of the circum - circle passing through all three vertices).
• The incenter is the intersection of the angle bisectors and is equidistant from the sides.
• The centroid is the intersection of the medians and is related to the balance point of the triangle.

We analyze each option:

• The mid - point of the medians is the centroid, not the incenter.
• The mid - point of the sides is not related to the incenter.
• The center of the circum - circle is the circumcenter, not the incenter.
• The incenter is defined as the center of the incircle (the circle tangent to all three sides of the triangle).

Answer:

d. Obtuse triangle

Second Question: