Question

Construction Draw an isosceles right triangle. Then construct the three angle bisectors of the triangle. (Lesson 6-3)

Short answer

Draw an isosceles right triangle, bisect the 90-degree angle and both 45-degree angles, and confirm they intersect at the incenter.

Step-by-step solution

Construct the Isosceles Right Triangle

Begin by drawing a horizontal line segment, which will be the base of the triangle. Ensure the segment is 5 cm long, or any other unit of measurement, maintaining uniformity. From one endpoint, draw a vertical line of the same length to form a right angle with the base. Connect the endpoints of these two line segments to form the hypotenuse, completing the isosceles right triangle. Verify that the triangle has two equal sides forming a right angle.

Identify the Angles of the Triangle

An isosceles right triangle has one right angle (90 degrees) and two equal angles (45 degrees each). Identify these angles at the vertices of your triangle.

Construct the Angle Bisector of the Right Angle

At the right angle vertex, use a compass to bisect the 90-degree angle. Place the compass at the right-angle vertex and draw an arc that crosses both sides of the angle. Using the same compass width, draw arcs from the points where the first arc crossed the angle sides to intersect with each other. Draw a straight line from the right-angle vertex through the intersection of the arcs. This line is the angle bisector.

Construct the Angle Bisectors of the 45-Degree Angles

Repeat the process for each of the 45-degree angles. Use the compass to draw an arc from the vertex to intersect the legs of the angle. Without changing the compass width, draw intersecting arcs from these points. Draw a line from the vertex through the intersection point to form the angle bisector.

Verify the Intersection Point

The intersection of the three angle bisectors is the incenter of the triangle. As a check, all three bisectors should meet at a single point inside the triangle, verifying correctness. This incenter equidistant from all three sides confirms the accuracy of the construction.

Key concepts

Isosceles right triangle

An isosceles right triangle is a special type of triangle that features a right angle, which is exactly 90 degrees, and two equal sides, known as the legs. Let's break it down further.
The two equal sides meet at the right angle and are equal in length. This gives rise to the name 'isosceles,' which indicates two sides being the same. The hypotenuse, being opposite to the right angle, is the longest side.

• The uniqueness of an isosceles right triangle lies in its angle measurements. Apart from one 90-degree angle, the other two angles measure 45 degrees each. This is because the total sum of angles in any triangle is always 180 degrees.
• Because of this equality, the trigonometry and geometry of isosceles right triangles allow for simpler calculations, often using the property of symmetry for problem-solving.

Constructing such a triangle involves ensuring that the two legs are of equal length and meet at a perfect right angle.

Angle bisector

The concept of an angle bisector is integral in understanding how to divide an angle into two equal parts. In any triangle, such as the isosceles right triangle, constructing an angle bisector involves a systematic approach using simple tools.

• Using a compass, start by placing its point at the angle's vertex. Then draw arcs across both sides of the angle. These arcs intersect the sides at two points each.
• Without adjusting the compass width, draw arcs from these intersection points to meet each other. The intersection point of these new arcs determines the path of the bisector from the vertex.
• The line passing through this intersection and the vertex is the angle bisector. It splits the original angle into two equal smaller angles.

Angle bisectors have a unique property - they converge at a single point within the triangle known as the incenter, which is equidistant from all sides of the triangle.

Triangle construction

Constructing a triangle is a fundamental aspect of geometry, and understanding the steps in this process can be quite enlightening. When constructing an isosceles right triangle, let's consider the following steps:

• Start by choosing a suitable length for the triangle's base. This line segment will act as one of the equal sides, usually drawn horizontally.
• From one endpoint of this base, draw another line perpendicular to it (forming a 90-degree angle) with a compass or protractor. Ensuring that it is of equal length as the base.
• Finally, connect the free end of the newly drawn line to the free end of the base to form the longest side, or the hypotenuse.

This process confirms both the angle and side properties needed to classify the figure as an isosceles right triangle. The careful sequence of actions aids in precision and accuracy, crucial for correct geometric constructions.

90-degree angle

The 90-degree angle is a cornerstone of geometry, especially in constructing shapes like the isosceles right triangle. A right angle is characterized by its perpendicularity between two lines, creating an "L" shape.

• In practical terms, to measure or draw a 90-degree angle, one commonly uses a protractor or a right-angle ruler to ensure perfect perpendicularity.
• In an isosceles right triangle, this right angle is adjoined by two lines of equal length, an indication of the special properties that distinguish this type of triangle.
• Understanding and accurately constructing a 90-degree angle is key not just in theoretical exercises but also in real-world applications seen in architecture, engineering, and various design fields.

Thus, the 90-degree angle serves as both a simple and vital element in geometry, with its applications stretching far beyond the classroom.