Every Rhombus With Four Right Angles Is A Square
Geometry is a branch of mathematics that deals with the study of shapes, sizes, and properties of figures. One of the basic figures in geometry is a rhombus. A rhombus is a four-sided polygon with opposite sides parallel and equal in length. In this article, we will discuss the property of rhombus that every rhombus with four right angles is a square.
What is a Square?
A square is a four-sided polygon with all sides equal in length and all angles equal to 90 degrees. It is a special type of rectangle where all sides are equal. The opposite sides of a square are parallel, and the diagonals bisect each other at right angles.
What is a Rhombus?
A rhombus is a four-sided polygon with opposite sides parallel and equal in length. It is also known as a diamond or a lozenge. The opposite angles of a rhombus are equal, and the diagonals bisect each other at right angles.
Proof that Every Rhombus with Four Right Angles is a Square
Let ABCD be a rhombus with four right angles, as shown in the figure below:
Since ABCD is a rhombus, all sides are equal. Let the length of each side be 'a'.
Draw diagonal AC which divides the rhombus into two congruent triangles, namely, △ABC and △ACD.
Now, we know that the sum of angles in a triangle is 180 degrees. Therefore, the sum of angles in △ABC is:
∠ABC + ∠BAC + ∠ACB = 180°
Since ∠BAC = ∠ACB (as ABCD is a rhombus), we can write:
2∠BAC + ∠ABC = 180°
As ∠ABC = 90° (as ABCD is a rhombus with four right angles), we get:
2∠BAC + 90° = 180°
2∠BAC = 90°
∠BAC = 45°
Similarly, we can prove that ∠DAC = 45°.
Now, consider the triangle △ADC. We know that:
∠ADC + ∠DAC + ∠ACD = 180°
As ∠DAC = 45° and ∠ACD = 90°, we get:
∠ADC + 45° + 90° = 180°
∠ADC = 45°
Therefore, ∠ACD = 45° (since AC bisects ∠ADC).
Now, consider the triangle △ABC. We know that:
∠ABC + ∠BAC + ∠ACB = 180°
As ∠BAC = 45° and ∠ACB = 90°, we get:
∠ABC + 45° + 90° = 180°
∠ABC = 45°
Therefore, ∠ACB = 45° (since AC bisects ∠ABC).
Thus, we have proved that △ACB is an isosceles right triangle with ∠ACB = 45°.
Using Pythagoras theorem, we can find the length of diagonal BD:
BD² = AB² + AD²
BD² = a² + a² (since AB = AD = a)
BD² = 2a²
BD = a√2
Since AC bisects BD, we have:
BC = CD = a/√2
Therefore, ABCD is a square as all sides are equal in length, and all angles are equal to 90 degrees.
Conclusion
From the above proof, we can conclude that every rhombus with four right angles is a square. This property of rhombus is essential in geometry and has several applications in different fields, such as architecture, engineering, and design. Understanding this property can help in solving various problems related to geometry.
So, next time you come across a rhombus with four right angles, remember that it is a square!
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