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

Attend live Mathematics sessions on Nagwa Classes to learn more about this topic from an expert teacher!

Draw the given figure and connect the points 𝐴𝐢𝐡𝐷. What does that figure represent?

03:01

Video Transcript

Draw the given figure and connect the points 𝐴𝐢𝐡𝐷. What does that figure represent?

Before connecting 𝐴𝐢𝐡𝐷, let’s label the intersection of line segments 𝐢𝐷 and 𝐴𝐡 point 𝐸. We see from the diagram that 𝐴𝐸 and 𝐸𝐡 have the same length and 𝐢𝐸 and 𝐸𝐷 have the same length. So we know that 𝐢𝐷 bisects line segment 𝐴𝐡 and 𝐴𝐡 bisects line segment 𝐢𝐷. We will now use a straightedge to connect the points 𝐴, 𝐢, 𝐡, and 𝐷 to form quadrilateral 𝐴𝐢𝐡𝐷.

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 𝐴𝐢, and the pink circle is centered at 𝐡 with radius 𝐡𝐢. We recognize this construction as the perpendicular bisector construction.

This means that line 𝐢𝐷 not only bisects line segment 𝐴𝐡, but it is the perpendicular bisector of line segment 𝐴𝐡. Therefore, 𝐢𝐡 bisects 𝐴𝐡 at right angles, which we can add to the diagram. Therefore, the diagonals of 𝐴𝐢𝐡𝐷 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 𝐴𝐸𝐢, 𝐡𝐸𝐢, 𝐡𝐸𝐷, 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 𝐴𝐢𝐡𝐷 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 𝐢𝐸 have the same length, we can’t prove that the diagonals are congruent. And therefore, we can’t claim that 𝐴𝐢𝐡𝐷 is a square.

Based on the evidence of the perpendicular diagonals and the congruent sides, we conclude that quadrilateral 𝐴𝐢𝐡𝐷 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

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy