Question

Construct the incircle of a given triangle \(\mathrm{ABC}\). The following sentences are the steps involved in the above construction. Arrange them in sequencial order from first to left. (a) Draw perpendicular \(\overline{\mathrm{IM}}\) from \(\mathrm{I}\) onto \(\overline{\mathrm{BC}}\). (b) Taking I as centre and IM as the radius, draw a circle. (c) Draw a \(\Delta \mathrm{ABC}\) with the given measurements. (d) Draw bisectors of two of the angles, say \(\angle \mathrm{B}\) and \(\angle \mathrm{C}\) to intersect at \(\mathrm{I}\). (1) dcab (2) cdab (3) cadb (4) dacb

Short answer

Question: Arrange the given steps in the correct sequential order to construct the incircle of triangle ABC: (a) draw a perpendicular from I onto BC, (b) draw a circle with center I and radius IM, (c) draw a triangle with given measurements, (d) draw angle bisectors for two of the angles. Answer: The correct sequential order is (c)(d)(a)(b).

Step-by-step solution

Draw the triangle with given measurements

First, draw triangle ABC with given measurements, as described in step (c).

Draw angle bisectors

Next, draw angle bisectors for two of the angles, say angle B and angle C, as described in step (d). These bisectors will intersect at point I.

Draw perpendicular from I onto BC

Now, draw a perpendicular line IM from point I onto the side BC, as described in step (a).

Construct the incircle

Finally, construct the incircle by taking point I as the center and IM as the radius, and draw a circle touching all three sides of triangle ABC, as described in step (b). Following our step-by-step solution, the correct order of steps is: (3) cadb.

Key concepts

Geometric Constructions

Geometric constructions are the foundation of many mathematical proofs and applications. They typically involve creating shapes, angles, and lines using a compass and straightedge. This practice dates back to ancient times and is still taught today to enhance spatial reasoning and problem-solving skills. In geometry, a construction must be precise and accurately replicable—no measuring devices are allowed beyond the compass and straightedge, which ensures that the construction follows rigorous mathematical principles.

In constructing the incircle of a triangle, you start with clear steps: drawing the triangle, constructing angle bisectors, drawing a perpendicular line to a side from the incenter, and finally drawing the circle itself. Each step in geometric construction should be carried out with precision since accuracy is key to obtaining the correct figure and ensuring that all students can both understand and replicate the process.

Triangle Incenter

The incenter of a triangle is the point where the angle bisectors of the triangle intersect. It is unique in that it is equidistant from all the sides of the triangle, making it the perfect center for constructing an incircle. This property is crucial for our construction exercise because it ensures that the incircle will be tangent to each side of the triangle. The incenter is one of the triangle's concurrent points, along with the centroid, circumcenter, and orthocenter.

The discovery of the incenter is a fundamental step in our given exercise; it is the point from which we draw the incircle. The incenter (I), being the center of the incircle, is the point used as the center for the circle that neatly fits within the confines of the triangle, touching all three sides.

Angle Bisectors

Angle bisectors are rays or line segments that divide an angle into two equal parts. In the context of a triangle, each angle can be bisected, and the point where two or more angle bisectors intersect is a significant centroid - in this case, the incenter. Understanding angle bisectors is essential because they not only help in locating the incenter but also influence the properties of the triangle. For instance, the incenter is always inside the triangle, no matter the type of triangle, because the angle bisectors of interior angles cannot intersect outside the triangle.

When constructing an incircle, you only need to bisect two angles; because of the properties of triangles, the third bisector will always meet at the same point. This creates a convenient efficiency in construction without compromising mathematical rigor. Angle bisectors are a vital element in various geometric proofs and present a visual representation of how geometry is both an art and a science.

Perpendicular Lines

Perpendicular lines are lines that intersect at a right angle (90 degrees). They play a critical role in geometric constructions, as they help to define and relate spaces and are foundational in creating shapes with right angles. In the construction of an incircle, once the intersection point I of the angle bisectors is determined, a perpendicular line from point I to one side of the triangle is drawn. This line, referred to as the perpendicular bisector, is vital because its length is equal to the radius of the incircle.

Perpendicular lines serve as a bridge between the angular aspect of a triangle and its sides. They allow us to create a link (IM in the given exercise) which is essential to constructing the incircle because the circle's radius has to be the same distance from all sides of the triangle. Understanding perpendicular lines not only facilitates this particular construction, it also deepens comprehension of the overall geometric structure, enriching the spatial reasoning vital to the study of geometry.