Video Transcript
Draw the given figure and
connect the points 𝐴𝐶𝐵𝐷. What does that figure
represent?
Before connecting 𝐴𝐶𝐵𝐷,
let’s label the intersection of line segments 𝐶𝐷 and 𝐴𝐵 point 𝐸. We see from the diagram that
𝐴𝐸 and 𝐸𝐵 have the same length and 𝐶𝐸 and 𝐸𝐷 have the same length. So we know that 𝐶𝐷 bisects
line segment 𝐴𝐵 and 𝐴𝐵 bisects line segment 𝐶𝐷. We will now use a straightedge
to connect the points 𝐴, 𝐶, 𝐵, and 𝐷 to form quadrilateral 𝐴𝐶𝐵𝐷.
To answer this question, we
must look for key features of the quadrilateral to determine what kind of
quadrilateral we are looking at.
We recall that a quadrilateral
can be classified as a trapezoid or a kite or a parallelogram, such as a
rectangle, rhombus, or square. We note that the intersecting
arcs in the diagram represent the intersection between two circles. The orange circle is centered
at 𝐴 with a radius 𝐴𝐶, and the pink circle is centered at 𝐵 with radius
𝐵𝐶. We recognize this construction
as the perpendicular bisector construction.
This means that line 𝐶𝐷 not
only bisects line segment 𝐴𝐵, but it is the perpendicular bisector of line
segment 𝐴𝐵. Therefore, 𝐶𝐵 bisects 𝐴𝐵 at
right angles, which we can add to the diagram. Therefore, the diagonals of
𝐴𝐶𝐵𝐷 are perpendicular. We know that the only
quadrilaterals that have perpendicular diagonals are kites, rhombuses, and
squares. Therefore, it is safe to
eliminate trapezoids and rectangles from our list of possibilities.
Based on the accumulated
evidence, it follows that triangles 𝐴𝐸𝐶, 𝐵𝐸𝐶, 𝐵𝐸𝐷, and 𝐴𝐸𝐷 are
congruent by the side-angle-side triangle congruence criterion, since all four
triangles have two congruent sides and the included angles are all right
angles. Thus, the corresponding sides
of the triangles are also congruent. This means all four sides of
quadrilateral 𝐴𝐶𝐵𝐷 are equal in length.
We know that rhombuses have
four congruent sides. And because a square is a
specific type of rhombus, squares also have the same property. We can eliminate kites as a
possibility, because they have two pairs of congruent sides. We recall that for a rhombus to
be a square, the diagonals must be congruent. Since we do not know if 𝐴𝐸
and 𝐶𝐸 have the same length, we can’t prove that the diagonals are
congruent. And therefore, we can’t claim
that 𝐴𝐶𝐵𝐷 is a square.
Based on the evidence of the
perpendicular diagonals and the congruent sides, we conclude that quadrilateral
𝐴𝐶𝐵𝐷 is a rhombus.
