Where can the perpendicular bisectors of the sides of a right triangle intersect?
I. inside the triangle
II. on the triangle
III. outside the triangle
(1 point)
Responses
I only
I only
II only
II only
I or II only
I or II only
I, II, or III
The perpendicular bisectors of the sides of a right triangle intersect at option II only, which is on the triangle.
The perpendicular bisectors of the sides of a right triangle can intersect at different locations depending on the triangle. The correct response would be "I, II, or III," indicating that the perpendicular bisectors can intersect inside the triangle, on the triangle, or outside the triangle.
The intersection of the perpendicular bisectors of the sides of a right triangle can be found by considering the properties of perpendicular bisectors and the specific shape of a right triangle.
First, let's define what perpendicular bisectors are. A perpendicular bisector is a line that cuts a segment into two equal parts (bisector) and is perpendicular to that segment. In the case of a right triangle, each side has a perpendicular bisector.
To determine where the perpendicular bisectors intersect, we need to understand the relationship between the sides of a right triangle. In a right triangle, one angle measures 90 degrees, and the other two angles are acute (less than 90 degrees). The side opposite the right angle is called the hypotenuse, while the other two sides are called the legs.
Now, let's consider the different possibilities for the intersection of the perpendicular bisectors:
I. Inside the Triangle: The perpendicular bisectors of the legs of a right triangle will intersect inside the triangle. This is because the legs of a right triangle are unequal in length, and the perpendicular bisectors will intersect at a point that is closer to the shorter leg.
II. On the Triangle: The perpendicular bisector of the hypotenuse of a right triangle will intersect on the triangle. This is because the hypotenuse is the longest side of the right triangle, and the perpendicular bisector will intersect at a point that is equidistant from the two legs.
III. Outside the Triangle: The perpendicular bisectors can never intersect outside the triangle. If they did, that would imply the existence of a point equidistant from the three sides, which is not possible in a right triangle.
Considering these possibilities, the correct answer is: I, II, or III, meaning that the perpendicular bisectors of the sides of a right triangle can intersect inside the triangle, on the triangle itself, or not at all (outside the triangle).