QCM : Triangle Center and Polygon Properties — 10 questions

Questions et réponses du QCM

1. What is a perpendicular bisector in geometry?

A line that is parallel to one side of a triangle
A line that divides a side into two equal segments at a right angle
A line that divides an angle into two equal parts
A line that connects the midpoints of two sides of a triangle

A line that divides a side into two equal segments at a right angle

Explication

A perpendicular bisector is a line that divides a side of a triangle into two equal segments at a 90° angle, passing through the midpoint of that side. It is used to find the circumcenter of the triangle.

2. What is the main property of perpendicular bisectors in a triangle?

They are parallel to each other.
They intersect at the incenter.
They are concurrent at the circumcenter.
They divide the triangle into two equal areas.

They are concurrent at the circumcenter.

Explication

Perpendicular bisectors of a triangle are always concurrent at the circumcenter, which is the point equidistant from all vertices, essential for constructing the circumscribed circle.

3. Where is the circumcenter located in a right triangle?

At the midpoint of the hypotenuse
Inside the triangle, equidistant from all vertices
At the intersection of the angle bisectors
Outside the triangle, on the extension of the altitude from the right angle

At the midpoint of the hypotenuse

Explication

In a right triangle, the circumcenter is located at the midpoint of the hypotenuse. This is a well-known property because the hypotenuse is the diameter of the circumcircle, and the center of the circle is at its midpoint.

4. Where does the circumcenter of a right triangle lie?

Inside the triangle.
On the hypotenuse.
Outside the triangle.
At the vertex of the right angle.

On the hypotenuse.

Explication

In a right triangle, the circumcenter lies exactly on the hypotenuse, at the midpoint of the hypotenuse, because it is equidistant from all three vertices.

5. What is the primary role of the angle bisector theorem in triangle geometry?

To relate the division of the opposite side to the lengths of the adjacent sides when an angle bisector is drawn
To establish that the diagonals of a parallelogram bisect each other
To prove that the medians of a triangle intersect at a single point
To determine the measure of an interior angle in a regular polygon

To relate the division of the opposite side to the lengths of the adjacent sides when an angle bisector is drawn

Explication

The angle bisector theorem specifically states that the bisector of an angle in a triangle divides the opposite side into segments proportional to the adjacent sides, establishing a key proportional relationship.

6. Which triangle center is the intersection point of the medians?

Orthocenter.
Centroid.
Incenter.
Circumcenter.

Centroid.

Explication

The centroid is found at the intersection of the medians, and it divides each median into a 2:1 ratio from vertex to midpoint, serving as the triangle's center of mass.

7. What is a key characteristic of the incenter of a triangle?

It is the point where the perpendicular bisectors intersect.
It is the intersection of the medians.
It is equidistant from all sides, being the center of the inscribed circle.
It always lies outside the triangle.

It is equidistant from all sides, being the center of the inscribed circle.

Explication

The incenter is the intersection of the angle bisectors and is equidistant from all sides, serving as the center of the inscribed circle, or incircle.

8. What is the purpose of constructing a perpendicular bisector in a triangle?

To find the incenter of the triangle.
To locate the centroid.
To find the circumcenter and construct the circumcircle.
To measure the triangle's height.

To find the circumcenter and construct the circumcircle.

Explication

Constructing a perpendicular bisector helps locate the circumcenter, which is used to draw the circumcircle passing through all three vertices of the triangle.

9. Which of the following correctly describes the orthocenter?

It is the intersection of the three perpendicular bisectors.
It is the incenter where the angle bisectors meet.
It is the point where the three medians intersect.
It is the point where the three altitudes intersect.

It is the point where the three altitudes intersect.

Explication

The orthocenter is the point of concurrency of the three altitudes of a triangle, which are perpendicular from each vertex to the opposite side or its extension.

10. In an obtuse triangle, where is the circumcenter located?

Inside the triangle.
On the hypotenuse.
Outside the triangle.
At a vertex.

Outside the triangle.

Explication

For an obtuse triangle, the circumcenter lies outside the triangle because it must be equidistant from all vertices, which extends beyond the triangle in such cases.

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Perpendicular bisectors — definition?

Lines dividing sides into two equal parts at 90°.

Perpendicular bisector — definition?

Divides a side into two equal parts at 90°.

Triangle centers — role?

Key points where special lines intersect, defining triangle properties.

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