Did you enjoy playing with Lego bricks as a child? Do you miss those days...or do you still pull out the Lego box every now and again and start building? People of all ages, not just young children, tend to find the tiny plastic bricks and what can be built with them fascinating.
That's why start-up company Titan Toys decided to manufacture a building block designed especially for teens. The company considers teens a powerful market force with plenty of their own money to spend on things other than the basics, since most don't have to pay for rent and food. To help him understand how teens might interact with "Buildz" blocks--and thus generate additional ideas for the product line, CEO Barry Thurston III hired several teen interns to test his prototypes. The interns' primary duties involved creating different structures with the bricks and then calculating the dimensions of what they build. Thurston explained to the teens that these calculations would help determine the number of Buildz blocks needed for each "kit" as well as other packaging requirements.
One teen, Olivia, was particularly drawn to a set of Buildz blocks consisting entirely of right triangles. The other interns avoided the triangular prototypes. Some assumed that triangles could be not be used to build anything interesting while others were unsure how to calculate the number of triangular pieces needed to create a design. Olivia felt differently, though. The slides below reveal a few of Olivia's creations.
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For her first creation, Olivia used 45-45 special right triangles to design a windmill.
She used the 1:1:\(\mathsf{ \sqrt {2} }\) ratio for 45-45 right triangles to calculate the windmill's diameter. Then Olivia combined two 30-60 right triangles to create a rectangle.
She used the 1:\(\mathsf{ \sqrt {3} }\):2 ratio for 30-60 right triangles to calculate the rectangle's length, width, and diagonal. Olivia saw great potential in this new shape. Next, Olivia designed a house using several special right triangles.
The rectangular base of the house was composed of 30-60 right triangles and the roof was a 45-45 right triangle. She used leg ratios to calculate dimensions of the house. Olivia then used special right triangles to build a see-saw.
One triangle served as a fulcrum and two others combined to make a rectangular seat. Olivia even used triangles to make tiny rectangular handles. Olivia used the leg rules for special right triangles to calculate dimensions of the fulcrum, seat, and handles. She noticed the seat and handles could also be used as a stretcher for medical transports. Thurston was impressed by Olivia's use of the right-triangle Buildz prototypes to model a neighborhood. Olivia included residential houses and commercial buildings in her kit design along with playgrounds and parks. She even used several windmills to provide clean energy for the neighborhood. |
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Titan Toys ended up using many of Olivia's designs to launch a campaign showing teens how they might use right triangle Buildz to build their geometry skills while creating amazing objects. In this lesson you, like Olivia, will discover how to build rectangles using various collections of triangles.
Question
What leg ratio did Olivia use to calculate the dimensions for structures that included 30-60 right triangles?
