Constructing an Isosceles Right-Angled Triangle with Hypotenuse AC5.5 cm

In this article, we will explore different methods to construct an isosceles right-angled triangle ABC where the hypotenuse AC 5.5 cm. This triangle has unique properties that make it a fundamental shape in geometry and trigonometry.

Method 1: Using a Perpendicular Bisector and Semicircle

To start, draw a horizontal line segment AC 5.5 cm. Next, construct the perpendicular bisector of AC, which will intersect it at its midpoint, say M. With M as the center and radius equal to AM MC 2.75 cm, draw a semicircle passing through both points A and C. The semicircle will intersect the perpendicular bisector at point B. The triangle ABC is the desired isosceles right-angled triangle. This construction ensures that AB BC, and the base angles ∠BAC and ∠BCA are both 45°.

Method 2: Using a Circle's Diameter and Radius

To construct an isosceles right-angled triangle, draw a circle with a diameter of 5.5 cm. Identify any point on the circumference, say B, by extending a line from the center of the circle, say O, to the circumference. The triangle ABC formed by connecting A, B, and C is the desired isosceles right-angled triangle. The length AO OC OB 2.75 cm, which is the radius of the circle. Additionally, since the inscribed angle in a semicircle is a right angle, ∠ABC 90°.

Method 3: Using Angle Bisectors

Another method involves drawing a line segment AC 5.5 cm. At the endpoints A and C, construct angles of 90°. Bisect these angles to form 45° angles. The intersection of the two 45° rays will be point B. The triangle ABC will then form the required isosceles right-angled triangle. This construction ensures that AB AC and the base angles ∠BAC and ∠BCA are both 45°.

Properties of Isosceles Right-Angled Triangles

Isosceles right-angled triangles have several unique properties. Since one angle is 90° and the other two angles are equal, each of these angles must be 45°. The sides opposite the 45° angles (i.e., AB and BC) are equal in length, and the side opposite the 90° angle (i.e., AC, the hypotenuse) is 5.5 cm. The relationship between the sides can be expressed using the Pythagorean theorem:

AB2 BC2 AC2

Given that AB and BC are equal (let's say each is 2.75 cm), the formula simplifies to:

2.752 2.752 5.52

This confirms the consistency and correctness of the construction.

Conclusion

There are multiple ways to construct an isosceles right-angled triangle with a given hypotenuse. Each method relies on fundamental geometric principles and properties of angles and circles. Whether using a perpendicular bisector and semicircle, a circle's diameter and radius, or angle bisectors, the resulting triangle will always have the desired properties of being isosceles and right-angled. Understanding these constructions is valuable in both mathematical and practical applications.