Prove that the diagonals of a rectangle are of equal length.
By BYJU'S Exam Prep
Updated on: September 25th, 2023
The diagonals of a rectangle are of equal length. Here, we need to demonstrate that a rectangle’s diagonals are equal. The two triangles (each with one diagonal as a side) are shown to be congruent using the characteristics of a rectangle. We can then prove that the diagonals of the rectangle are equal by using the congruent sections of congruent triangles, which are equivalent.
Table of content
Length of Diagonal of a Rectangle
To prove that the diagonals of a rectangle are equal, we will use the characteristics of a rectangle and the congruence of triangles.
Solution:
Given that ABCD is a rectangle, we know that AC and BD are its diagonals.
Consider triangles ABC and BCD. Angle B is a common angle between the two triangles, and BC is a common side.
From the properties of a rectangle, we know that opposite sides are congruent. Therefore, AB = CD.
Applying the SAS (Side-Angle-Side) congruence criterion, we have angle B as a common angle, BC as a common side, and AB = CD as the other pair of congruent sides.
By SAS congruency, triangle ABC is congruent to triangle BCD.
Using the CPCT (Corresponding Parts of Congruent Triangles) theorem, we can conclude that the corresponding sides of congruent triangles are congruent.
Applying CPCT, we have AC as the corresponding side to AB in triangle ABC, and BD as the corresponding side to CD in triangle BCD.
Since AB = CD and triangle ABC is congruent to triangle BCD, it follows that AC = BD.
Therefore, it is successfully proven that the diagonals AC and BD of the rectangle ABCD are of equal length.
Summary:
Prove that the diagonals of a rectangle are of equal length.
Using the SAS (Side-Angle-Side) congruence criterion to establish that triangles ABC and BCD are congruent, and then applying the CPCT (Corresponding Parts of Congruent Triangles) theorem to conclude that AC is equal to BD. Therefore, the diagonals of the rectangle ABCD are indeed of equal length.
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