Chapter 5: Triangle Constructions (part 1: 3 sides)

Auteur :mbader97, MathPatty1, Melissa Small

Chapter 5 Packet Overview The goals of this construction packet are: 1) Learn how to copy an angle using compass tools and understand why this method works. 2) Be able to copy triangles by copying segments and/or angles. 3) Explore and discover which "S" and "A" congruence combinations lead to triangle congruence. 4) Understand why certain "S" and/or "A" combinations do NOT guarantee triangle congruence.

#1) Testing for Side-Side-Side ("SSS") Congruence

Goal: Let's try to copy a triangle by copying all three side lengths. Step 1: Copy one side length. - Use the "COMPASS" tool to make a circle with radius = AB, centered at A'. - Use the "POINT" tool to pick any point on the circle. Rename (right click) this point as B'. - Use the "SEGMENT" tool to connect A' and B'. - Now you have a side whose length will always equal AB. Step 2: Copy a second side length. - Create a circle with radius = AC, centered at A'. Step 3: Copy the third side length. - Create a circle with radius = BC, centered at B'. - Use the "INTERSECT" tool to place a point where the two circles intersect. - Rename this point as C'. - Because C' lies on both circles, A'C' = AC and B'C' = BC. Step 4: "lighten" the arc marks. - Click on any arc or segment. A color & line menu should appear in the top right. You can use it to change the color, thickness, and/or dotted-ness of your work. Now, the three sides should stand out visually, but the circles are still visible as a way to show your work.

Testing "SSS"

Move points A, B, and C around to produce different types of triangles. As you move these points, triangle A'B'C' will also change. Using inductive reasoning (testing many different possible triangles), do you think "SSS" congruence is true? In other words, if two triangles have three sets of congruent corresponding sides, will they always be congruent triangles?

#2) When Can Three Side Lengths Form a Triangle?

Play around with the sliders in the window below. a) Look for combinations of side lengths that will form a triangle b) Look for combinations that cannot form a triangle. c) Look for a combination of side lengths that allow all three vertices to connect, but will NOT form a triangle. Leave this combination visible in your submission.

In order for three side lengths to form a triangle, what must be the relationship between the longest side and the two other sides?

#3a) Use SSS to Copy a Triangle into a Kite

In the window below, create a kite by reflecting the triangle over its left side. 1) Create a circle with center P and radius PQ. 2) Create a circle with center R and radius RQ. 3) Find and label point Q' so that triangle PRQ will be congruent to triangle PRQ'. 4) Use the "segment" tool to add the sides.

Construct three copies of the triangle below, reflecting it over each of its sides. This will require 6 circles: each side of the triangle becomes the radius of a circle, centered at each endpoint. It may help to lighten your circles and add the new segments as you go. See page 44 of the printed construction packet for a visual of what the result should look like.

How many kites are in the new construction above?

#3b) Use SSS to Copy a Triangle into a Parallelogram

Create a parallelogram by rotating a triangle around the midpoint of its left side. 1) Create a circle with center P and radius RQ. 2) Create a circle with center R and radius PQ. 3) Find and label point Q' so that triangle PQR will be congruent to triangle RQ'P. 4) Use the "segment" tool to add the sides.

Construct three copies of the triangle below, rotating it around the midpoint of each of its sides. This will require 3 circles: each side of the triangle becomes the radius of a circle, centered at the opposite endpoint. It may help to lighten your circles and add the new segments as you go. See page 45 of the printed construction packet for a visual of what the result should look like.

How many parallelograms are in the new construction above?

Consider your construction above as one large triangle with the original triangle inside of it. Each side of the original triangle (in red) is called a midsegment of the larger triangle, because each red side connects the midpoints of two sides of the larger triangle. How does the length of each midsegment compare with the length of the side it is parallel to?

How does the area of the inner triangle (red) compare to the area of the outer triangle?

Chapter 5: Triangle Constructions (part 1: 3 sides)

Testing "SSS"

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