Question Video: Linking the Construction of the Perpendicular Bisector of a Line Segment with the Properties of a Rhombus | Nagwa Question Video: Linking the Construction of the Perpendicular Bisector of a Line Segment with the Properties of a Rhombus | Nagwa

# Question Video: Linking the Construction of the Perpendicular Bisector of a Line Segment with the Properties of a Rhombus Mathematics • First Year of Preparatory School

## Join Nagwa Classes

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

03:01

### Video Transcript

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

Before connecting π΄πΆπ΅π·, letβs label the intersection of line segments πΆπ· and π΄π΅ point πΈ. We see from the diagram that π΄πΈ and πΈπ΅ have the same length and πΆπΈ and πΈπ· have the same length. So we know that πΆπ· bisects line segment π΄π΅ and π΄π΅ bisects line segment πΆπ·. We will now use a straightedge to connect the points π΄, πΆ, π΅, and π· to form quadrilateral π΄πΆπ΅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΄πΆ, and the pink circle is centered at π΅ with radius π΅πΆ. We recognize this construction as the perpendicular bisector construction.

This means that line πΆπ· not only bisects line segment π΄π΅, but it is the perpendicular bisector of line segment π΄π΅. Therefore, πΆπ΅ bisects π΄π΅ at right angles, which we can add to the diagram. Therefore, the diagonals of π΄πΆπ΅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΄πΈπΆ, π΅πΈπΆ, π΅πΈπ·, and π΄πΈπ· 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΄πΆπ΅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΄πΈ and πΆπΈ have the same length, we canβt prove that the diagonals are congruent. And therefore, we canβt claim that π΄πΆπ΅π· is a square.

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

## Join Nagwa Classes

Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!

• Interactive Sessions
• Chat & Messaging
• Realistic Exam Questions