In this Cyberchase video segment, Matt and Jackie must figure out how to construct rectangular lids that match the size of two boxes precisely. They enlist the carpentry skills of one of the Three Little Pigs, who shows them how to use different tools to accurately measure the rectangular lids. For the first box, Jackie finds that she can easily trace the box to make a lid that fits perfectly. The second box cannot be traced, so Matt and Jackie decide to measure the sides of the box and then use the measurements to cut a rectangle of equal size out of a piece of Herculanium.
Replicating a shape can be done in different ways, depending on the characteristics of the shape and the tools available. Tracing the outline of a shape is an easy way to know that you have created an exact match. You can use this technique with a pattern to cut fabric for a project such as a pillow or dress. Using a pattern makes it easy to create multiple objects without using measuring tools each time. You can use similar tools like a stencil or a cookie cutter to create duplicate shapes. For larger shapes, however, these techniques may be insufficient.
Sometimes it is necessary to take measurements to create an exact match. To start, it is necessary to note the characteristics of the shape, such as the lengths of the sides and the measure of the angles between adjacent sides. It is possible to use width and length measurements to create a parallelogram that is not a rectangle. Therefore, the fact that each of the four angles of a rectangle is ninety degrees is an important detail needed to produce an exact match.
It is worth noting that all rectangles are parallelograms, but not all parallelograms are rectangles. Squares are rectangles (and parallelograms), but they also have the special feature of four congruent sides and four 90-degree angles. There are several tools that can be used to create right angles: two include a carpenter's square and a protractor. Mathematical formulas provide another way to verify that an angle is a right angle. If you create a triangle by joining the two sides that comprise the angle, you can check whether the sum of the squares of the two sides is equal to the square of the hypotenuse (the side opposite the right angle). If it is, then the angle is a right angle. This is the converse of Pythagorean's Theorem.
The need for taking measurements arises often in daily life. For example, it is not practical to trace a large window and then bring that piece of paper into the store to buy blinds. Instead, it is more efficient to measure the exact width and height of the window and give these measurements to a store employee before they cut the blinds.
ZIFF: This is all the Herculanium I could find!
JACKIE: That's great! Thanks, Ziff.
MATT: Looks like enough for the lids.
MATT: But...how are we gonna make a lid out of this weird shape? It's gotta fit the box exactly.
JACKIE: I have an idea. Lay it on the ground and put the box on top.
JACKIE: See? Now we can trace around the box and get the exact shape.
ZIFF: Here, use my carpenter's pencil.
ZIFF: I don't mean to be critical, but that doesn't look like it's going to fit the box.
JACKIE: You're right. What happened?
MATT: Maybe the box moved when you traced it. I'll hold it for you this time.
ZIFF: Now we're talkin'!
MATT: How are we gonna cut it out?
ZIFF: Wanna try my new laser cutter? And you better put on these safety goggles.
JACKIE: Rrrg, arrg...the big box is totally stuck! There's no way we can trace the box like we did last time.
ZIFF: Then how are we going to make a lid that fits perfectly?
MATT: Everybody, chill! Look at the box. It's a rectangle. So all we have to do to make a rectangle for the lid. Like this. See?
JACKIE: That's true, but we should measure the sides. We don't have any Herculanium to spare, so we can't make a mistake.
ZIFF: Here, use my tape measure.
JACKIE: Awesome! Thanks, Ziff! Matt, I'll measure - you write down the numbers. Sixteen cyber-inches on this side. And this side, too. The opposite sides are equal. Eighteen cyber-inches across the top and the bottom.
MATT: Got it! Let's draw the real deal!
ZIFF: Ruler, anyone?
JACKIE: Ziff, you are amazing!
MATT: Go for it, Jax.
JACKIE: Sixteen cyber-inches up the side. And 18 across! I'll just finish it off. Got it.
MATT: Uh, Jax? Your rectangle is kinda leaning.
JACKIE: Yeah, but why...? I measured right...and my lines are straight.
ZIFF: Here's the problem. Your drawing doesn't have square corners. A rectangle has to have square corners.
MATT: What is that thing?
ZIFF: It's called a Carpenter's Square, 'cause it's got a square corner. See?
MATT: Hey! Is that corner square?
ZIFF: Yep. Square as can be!
JACKIE: That gives me an idea! May I have that?
JACKIE: Since we know this corner is square, I'll start the rectangle here. Sixteen cyber-inches up this side and 18 across the bottom. Ummmm....how do I use this to make the rest of the rectangle?
ZIFF: I saw my brothers do this once. Put it here to make the square corner. Just line up the straight side of the carpenter's square with the straight side of your material.
JACKIE: Got it! Eighteen cyber-inches across the top. And 16 up this side! And we've got our rectangle!
MATT: Home run, team! Let's cut to fit.
ZIFF: Laser cutter!
JACKIE: A perfect fit! We're two for two!
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