Quadrilaterals · Lesson 9 of 13
Building Quadrilaterals
“Use geoboards and joined triangles to build, predict and justify different quadrilateral types.”
• Use diagonal conditions on a geoboard. • Explain the effect of extending both ends of a diagonal. • Join congruent triangles in different ways. • Justify the resulting quadrilateral from its boundary.
Two identical triangle cutouts can make more than one four-sided figure. The result depends on which edges you join and how the two triangles face each other. Similarly, moving the endpoints of two diagonals can change a square into a different shape while preserving some properties. These activities give you a chance to use the definitions as tools: make a prediction, build the figure and explain which conditions still hold.
Use cutouts or a dot grid if you do not have a geoboard. A successful answer includes a reason, not just the name of a shape that your drawing resembles.
4.5 Playing with Quadrilaterals
Geoboard Activity
Place two equal perpendicular diagonal segments so that they share the same midpoint. Join their endpoints around the outside. All three square conditions are present: equal length, mutual bisection and perpendicularity.
The common midpoint matters. Equal perpendicular segments placed off-centre would not guarantee a square. Here we keep the midpoint fixed as we make changes.
Joining Triangles
When two triangles are joined along a whole edge with interiors on opposite sides, the shared edge becomes internal. Count only the four unjoined edges when checking the outer quadrilateral.
For two congruent equilateral triangles, join one 8 cm side of each. Every outer edge is also 8 cm, so the boundary is a rhombus. At each end of the shared edge, two 60° angles combine to 120°. The two other corners remain 60°.
Problem
Two equilateral triangles of side 4 cm are joined along one side. Find the boundary sides and angles.
Now use two isosceles triangles with sides 8 cm, 8 cm and 6 cm. Joining their 6 cm bases places four 8 cm sides on the boundary, so it produces a rhombus.
Another arrangement joins an 8 cm side and places the second triangle as a half-turn of the first around the midpoint of that side. The outer sides alternate 8 cm and 6 cm. Both pairs of opposite sides are equal, and the corresponding directions are parallel, giving a parallelogram.
If instead you reflect across a joined side, you put equal lengths next to one another. This is a different matching arrangement. Track the actual side labels instead of assuming that the triangle type fixes the result.
Problem
Compare joining the 6 cm bases of two 8–8–6 triangles with half-turn joining along an 8 cm side.
Finally use two scalene triangles with sides 6 cm, 9 cm and 12 cm. Reflecting one across a shared 12 cm side makes two pairs of equal adjacent boundary edges: 6 with 6, and 9 with 9. That shape leads into the kite lesson.
Alternatively, make a half-turn join along a corresponding side. The two other lengths appear as opposite boundary pairs, producing a parallelogram. With any arrangement, check that the outer boundary has four distinct corners and does not cross itself.
Problem
Two 6–9–12 triangles share their 12 cm sides. How can the arrangement change the outer side pattern?
Quiz
Equal diagonals that bisect each other at 90° form a:
Unequal diagonals that bisect each other at 90° point to a:
Two congruent triangles are joined along a whole matching side. What determines the outer figure?
Two equilateral triangles joined along a side produce an outer quadrilateral with:
A geoboard activity is useful mainly because it lets you:
Practice Problems
- Why do equal perpendicular diagonals that bisect each other give a square?
- A square has 4 cm diagonals. One is extended by 2 cm at each end. What are the new diagonal lengths and shape?
- Join two equilateral triangles of side 6 cm. Find all boundary sides and angles.
- Join two 8–8–6 triangles along their 6 cm bases. What outer figure results?
- For two 6–9–12 triangles joined along the 12 cm side, explain how to obtain a kite or a parallelogram.
1. Equality and mutual bisection give a rectangle. 2. Perpendicularity makes its four small triangles congruent, giving four equal boundary sides. 3. A rectangle with four equal sides is a square.
1. The extended diagonal becomes 4 + 2 + 2 = 8 cm; the other stays 4 cm. 2. They remain perpendicular and mutually bisecting, so the figure is a rhombus. 3. Unequal diagonals show it is not a square.
1. All four outer sides are 6 cm. 2. Two corners are 60°, and the two shared-endpoint corners are 120°. 3. The result is a rhombus.
1. The bases become an internal diagonal. 2. The four exposed sides are all 8 cm. 3. Four equal sides give a rhombus.
1. Reflect the second triangle across the joined side to place 6 cm sides together at one endpoint and 9 cm sides together at the other. This gives a kite. 2. Half-turn the second triangle around the midpoint instead: the exposed side lengths alternate 6 and 9, with matching sides opposite. 3. That arrangement gives a parallelogram. Identify the orientation as well as the joined side.
Key Takeaways
• A square construction needs equal, perpendicular, mutually bisecting diagonals. • Extending both ends equally preserves a diagonal’s midpoint. • Ignore the internal joined edge when classifying the outer boundary. • Joining equilateral triangles gives a rhombus with 60° and 120° angles. • The joined side and the orientation both affect the result. • Use side and angle evidence to justify a construction.