Back To CourseGeometry: High School
15 chapters | 160 lessons
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Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.
It's time to build. Geometry is awesome because you get to draw pictures, play connect the dots and other fun stuff, and it's legitimate mathematics. Here, we're going to practice constructing triangles. This involves using a compass and a straight edge (or a ruler) to build three-sided shapes.
To do this, we're heading off to a job site where a new neighborhood of triangle-shaped homes are ready to build. So get out your compass, ruler, pencil and a hard hat, and let's make some triangles! Okay, maybe you don't need a hard hat.
Here's triangle ABC:
We're working on a neighborhood with four different models, and this is our first one. We want to copy it, which means making a congruent triangle. Thank goodness we live in two-dimensional geometry world, or we'd probably need to know things about plumbing and electrical work.
Let's start with a point. Let's call it D. Now, draw a ray using the ruler from D to form our base, or foundation of the house (see video starting at 01:00 to see these actions). Next, take the compass and measure the distance from A to C. Use this width to draw an arc that hits our ray. Where it hits is point F, the equivalent of point C on the model.
Next, use the compass to measure the distance from A to B. Again, use this to draw an arc around where the top of the new triangle should be. Don't add a point yet. We're not sure exactly where the top of our house will be.
Let's measure C to B and draw another arc, this time from point F. Where these arcs meet is our final point, point E. Now, just connect D to E and F to E, and we have a congruent triangle.
That's the first model. Time for the next. Oh no! There's no model house to copy. This time, there's just these parts below: two sides and an included angle. We're given sides AB and AC, as well as angle A.
Well, this triangle isn't going to build itself. Let's get started. Let's start with point A. As before, draw a ray from A (see video starting at 02:05 for these actions). Then, measure AB with the compass, add an arc that hits the ray, and add point B.
Now draw a small arc on angle A. Then, keeping the same width on the compass, draw a similar arc on point A of our triangle. Back on angle A, match the compass to the points where the arc hits the angle. Then match this on the new arc we drew with, yep, another arc. With a ruler, draw a ray from point A through where the arcs meet.
Next, use the compass to measure AC, then draw an arc from A on the new triangle that hits the ray. This is our new AC. Finally, connect C to B with the ruler, and we did it! Another successful construction project.
Our neighborhood would be boring with just two models. Let's add a third. Oh man, they keep challenging us. Now we have two angles and an included side below. We have side AB, and angle A and angle B. Okay, compass? Check. Ruler? Check. Nail gun? No? Yeah, that's probably for the best.
Let's start, as always, with point A (see video starting at 03:17 to see these actions). Again, draw a ray, then measure AB with the compass, add an arc on the ray, and where they meet is B. That's our new AB.
Now we copy that trick with the angle from the last job. On angle A, draw an arc. Then draw a congruent arc on the new A. Now use the compass to measure where the arc hits angle A, then match that on the new triangle. Draw a ray from A through that point.
Now we do the same thing with angle B. Draw an arc. Then another on new point B. Then measure angle B and use that on the new arc. Add a ray from B through that point, and we have a triangle!
It's worth noting that while these two angles and a side worked just fine, not any pair of angles will work. If our angles looked like below, both right angles, or a right angle and an obtuse angle, well, we couldn't make the sides meet. All of the angles in a triangle must add up to 180 degrees. So, if two angles add up to 180 or more, we can't draw a triangle. It's like a house with no roof. That's all well and good until it rains.
Let's do one more model in our neighborhood. We're feeling pretty confident at this point, so let's try making an equilateral triangle when we're given just one side. This is like building a house based on one two-by-four. What would Bob the Builder say? Can we fix it? Yes, we can!
So, here's side AB:
Let's draw new point A. This time, let's jump right to measuring AB with the compass (see video starting at 04:53 for these actions). In an equilateral triangle, all sides are the same. So, this is the only measurement we need.
Draw two arcs from new point A. One up here and one out here. On the one over here, add new point B. Now move the compass to point B and draw another arc up here. Where these arcs meet is point C. Let's bust out our ruler, connect the dots, and we have our equilateral triangle!
What a busy day. We built four triangle houses with nothing more than a compass and a ruler. First, we made a congruent triangle. This involved copying the sides of the original. Next, we constructed a triangle based on two sides and an angle. This was a bit more hands-on, but we did it. Third, we made a triangle using two angles and a side. That used the same skills as the previous one. Finally, we made an equilateral triangle, or a triangle with three equal sides, using just a single given side. And that's constructing triangles!
You should be able to construct triangles with a compass and a ruler under four different scenarios after watching this video lesson.
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Back To CourseGeometry: High School
15 chapters | 160 lessons