To prop up a pole, a guy wire is placed 48' from the base of the 20' pole. Another stabilizing pole is places as shown as the wire's midpoint. How long is the second pole?
We notice that 48' here is the distance between a point (where the wire touches the ground) and a line (the pole). The distance between a point and a line is defined as the perpendicular distance. Thus 20' and 48' are in fact the legs of a right triangle.
The question is actually asking for the median to the hypotenuse of this right triangle. Now there's a theorem stated in some Geometry texts (but not the U of Chicago text) that the median to the hypotenuse is half the length of that hypotenuse. This is ultimately proved by rotating the triangle 180 degrees, centered at the midpoint of the hypotenuse. Then a rectangle is formed, and then the result (median is half the hypotenuse) follows from two known theorems -- that the diagonals of a rectangle are congruent, and that the diagonals of a parallelogram (which a rectangle is) bisect each other.
And so all that remains is to find the hypotenuse, using the Pythagorean Theorem:
a^2 + b^2 = c^2
20^2 + 48^2 = c^2
400 + 2304 = c^2
2704 = c^2
c = 52
and then taking half of it to find the median -- 26'. Therefore the desired length is 26 feet -- and of course, today's date is the 26th.
(By the way, when I cut and paste last year's Lesson 3-1 post, I notice that the Pappas question that day was on the altitude to the hypotenuse, while today's question is on the median to the hypotenuse.)
Lesson 3-1 of the U of Chicago text is called "Angles and Their Measures." This is what I wrote last year about today's lesson:
I don't have much to say about the book's treatment of angles. This text is unusual in that it includes a zero angle -- an angle measured zero degrees. Then again, in Common Core we may need to discuss the rotation of zero degrees -- the identity function.
The key to angle measure is what this text calls the Angle Measure Postulate -- so this is the second major postulate included on this blog. Many texts call this the Protractor Postulate -- since protractors measure angles the same way that rulers measure length for the Ruler Postulate. The last part of the postulate, the Angle Addition Property, is often called the Angle Addition Postulate. Notice that unlike the Segment Addition Postulate -- which this text calls the Betweenness Theorem -- the text makes no attempt to prove the Angle Addition Postulate the same way. Notice that Dr. Franklin Mason's Protractor Postulate -- in his Section 1-6, has angle measures going up to 360 rather than 180.
So let me include a few more things in this post. First, here's another relevant video from Square One TV -- the "Angle Dance":
By the way, three years ago I taught angles to my seventh graders.
No comments:
Post a Comment