Geometry Asn Questions (Always, Sometimes, Never)

Explanation
A right triangle cannot be equilateral because an equilateral triangle has all sides equal in length, whereas a right triangle has one angle measuring 90 degrees. In an equilateral triangle, all angles are 60 degrees. Therefore, a right triangle cannot have all sides equal in length and cannot be equilateral.

Explanation
If a triangle has an obtuse angle, then its orthocenter is always outside the triangle. This is because the orthocenter is the point of intersection of the altitudes of a triangle, and an obtuse triangle has one of its altitudes outside the triangle. Therefore, the orthocenter cannot be inside the triangle and is always located outside.

Explanation
The perpendicular bisectors of the two diagonals in a rectangle are sometimes perpendicular. This is because in a rectangle, the diagonals are equal in length and bisect each other. The perpendicular bisectors of these diagonals will intersect at the center of the rectangle, creating four right angles. However, in some cases, the rectangle may be a square, where all angles are right angles, making the perpendicular bisectors of the diagonals also perpendicular. Therefore, the statement is sometimes true, depending on whether the rectangle is a square or not.

Explanation
If an equilateral triangle shares 2 sides with a parallelogram, then it is a rhombus. This is always true because an equilateral triangle has all sides equal in length, and a parallelogram has opposite sides equal in length. If two sides of the equilateral triangle are also sides of the parallelogram, then all the sides of the parallelogram must be equal in length, making it a rhombus.

Explanation
An isosceles triangle has two sides of equal length. In this type of triangle, the base is always shorter than the sum of the two legs. This is because the two legs are equal in length, so their sum will always be greater than the base. Therefore, the correct answer is "Never."

Explanation
The segments connecting two adjacent endpoints of the diameters of a single circle are congruent because they are radii of the same circle. A radius is a line segment with the same length as any other radius of the same circle. Therefore, regardless of the size or position of the circle, the segments connecting the endpoints of the diameters will always be congruent.

Explanation
The exterior angle of a regular polygon can be less than the interior angle in certain cases. This occurs when the regular polygon has more than 3 sides. For example, in a regular pentagon, each interior angle measures 108 degrees, while each exterior angle measures 72 degrees. Therefore, the statement is true in some situations, but not always.

Explanation
The centroid and incenter of a triangle are not always the same point. The centroid is the point of intersection of the medians of a triangle, while the incenter is the point of intersection of the angle bisectors. In some special cases, such as an equilateral triangle, the centroid and incenter coincide, but in general, they are different points.

Explanation
When an angle is bisected, it means that it is divided into two equal parts. In the case of an angle inscribed in a circle, if it is bisected, the two equal parts will intersect at the center of the circle. However, not all angles inscribed in a circle are bisected, so the bisection ray does not always pass through the center. Therefore, the correct answer is "Sometimes".

Explanation
In a cube, the longest distance is the diagonal that passes through the center of the cube and connects two opposite corners. This diagonal can be found using the Pythagorean theorem, which states that the square of the length of the diagonal is equal to the sum of the squares of the lengths of the sides. In a cube, all sides are equal, so if we let the length of a side be "s", then the length of the diagonal (d) can be found by d^2 = s^2 + s^2 + s^2 = 3s^2. Taking the square root of both sides, we get d = √(3s^2) = √3s. Therefore, the longest distance in a cube is always √3 times its side length.

Explanation
I did not say that the trisection points are on the legs of the isoceles triangle.

Explanation
When the midpoints of all the sides of an equilateral triangle are connected, the resulting triangle is also an equilateral triangle. This new triangle has side lengths that are half the length of the original triangle. When this process is repeated three more times, the resulting triangle will have side lengths that are 1/16th the length of the original triangle. Since the area of a triangle is proportional to the square of its side length, the area of the final triangle will be (1/16)^2 = 1/256 the area of the original triangle. Therefore, the statement "the area of the final triangle is 1/256 the original area" is always true.

Explanation
If AB is the diameter of circle O and the line segment AB is tangent to the circle at its midpoint, then the line segment AB is perpendicular to the radius of the circle at its midpoint. This is because the tangent to a circle is always perpendicular to the radius at the point of tangency. Therefore, triangle OAB is a right triangle, with angle OAB equal to 90 degrees. Thus, the statement is always true.