### Video Transcript

Draw the given figure and
connect the points π΄πΆπ΅π·. What does that figure
represent?

Before connecting π΄πΆπ΅π·,
letβs label the intersection of line segments πΆπ· and π΄π΅ point πΈ. We see from the diagram that
π΄πΈ and πΈπ΅ have the same length and πΆπΈ and πΈπ· have the same length. So we know that πΆπ· bisects
line segment π΄π΅ and π΄π΅ bisects line segment πΆπ·. We will now use a straightedge
to connect the points π΄, πΆ, π΅, and π· to form quadrilateral π΄πΆπ΅π·.

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 π΄ with a radius π΄πΆ, and the pink circle is centered at π΅ with radius
π΅πΆ. We recognize this construction
as the perpendicular bisector construction.

This means that line πΆπ· not
only bisects line segment π΄π΅, but it is the perpendicular bisector of line
segment π΄π΅. Therefore, πΆπ΅ bisects π΄π΅ at
right angles, which we can add to the diagram. Therefore, the diagonals of
π΄πΆπ΅π· 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 π΄πΈπΆ, π΅πΈπΆ, π΅πΈπ·, and π΄πΈπ· 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 π΄πΆπ΅π· 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 π΄πΈ
and πΆπΈ have the same length, we canβt prove that the diagonals are
congruent. And therefore, we canβt claim
that π΄πΆπ΅π· is a square.

Based on the evidence of the
perpendicular diagonals and the congruent sides, we conclude that quadrilateral
π΄πΆπ΅π· is a rhombus.