# Isosceles Tutorial: Constructing Parallel Lines

A common task in geometry is to construct a line through a point that is parallel to a given line. Before starting this tutorial, make sure you have Shows Intersections turned on the Canvas section of Settings.

1. We are given a line (AB), and a point C.

2. Draw a line from C that passes through line AB. This is called a transversal.

3. Tap the Arc tool. The compass should appear in the center of the screen.

4. Drag the center handle of the compass to move it to the intersection of the two lines. Drag the compass arm to rotate it to the position shown.

5. Draw an arc across line AB, then move the compass handle to point C.

6. Draw a similar arc that would intersect the transversal. The arcs should now look like this with a different tool selected to hide the compass:

7. Now we need the intersection of the transversal and this last arc. Tap and hold on the transversal to select it, then tap it to open the menu if it isn’t already. Choose Info, then turn on Extends.

8. Move the compass to the intersection of the transversal and the first arc. Drag the outer handle to resize it. The outer handle should be at the intersection of line AB and the arc.

9. Move the compass to the intersection of the transversal and the second arc. Now draw a third arc across the second arc. This marks the second point of the parallel line.

10. Draw a line with the Line tool between point C and this latest point. If you follow the steps above to extend the line, the diagram will finally look like this:

# Problem of the Day: 1/31/13

In the diagram below, the shortest paths from A to B along the gridlines are 6 units long. How many of these paths are there?

Solution to yesterday’s problem:

The trick here is that all of the exponents have two as a base. So we have to find a way to take out all the bases and leave only the exponents in our equation. Here’s the equation again:

$2^{2n+3}=\frac{2^{n-2}}{2^{2n-2}}$

Recall that when you divide two exponents with the same base, the result is the base taken to the difference of the two exponents. For example, $\frac{x^5}{x^2}=x^{5-2}=x^3$. Here, we can subtract the two exponents as well:

$latex\frac{2^{n-2}}{2^{2n-2}}=2^{(n-2)-(2n-2)}=2^{-n}$

So our new equation is $2^{2n+3}=2^{-n}$

In order for the two sides of this equation to be equal, both exponents have to be equal. So here’s where we take out the bases:

$2n+3=-n$

$3n=-3$

$n=-1$

So n = -1.

# Problem of the Day: 1/30/13

If $2^{2n+3}=\frac{2^{n-2}}{2^{2n-2}}$, what is n?

(This equation has been corrected from yesterday.)

Solution to yesterday’s problem:

Here’s the diagram again, with some small additions:

Notice that each of the four triangles is a 30-60-90. This means we can find the side lengths of all the sides given one of the sides. Recall that the short side is n, the hypotenuse is 2n, and the long side is n times the square root of 2. We know that 2n = 8 cm, so the short side must be 4cm. Now we can make another interesting addition to the diagram considering that a tangent line is perpendicular to a circle’s radius:

Now we have another 30-60-90 triangle because one of the angles is the same, and one of the angles is 90°. Knowing that the hypotenuse is 4 cm, the short side is 2 cm, and the long side is $2\sqrt{3}$ cm.

# Isosceles Tutorial: Bisecting a Line Traditionally

Many geometry techniques require you to bisect a line. In this tutorial you will learn how to find the midpoint and perpendicular bisector of a line. First, ensure that Shows Intersections is turned on in the Canvas pane of the Settings.

1. Create a point to start the line.

2. With the Line Tool selected, drag from the point to create the line to bisect. (Obviously, you will not need to do this if you already have a line to bisect!)

3. Select the Arc tool. The compass should appear in the middle of the screen.

4. Move the compass to point A by dragging the center handle. Drag the compass arm to rotate it to the position shown.

5. First tap the pencil icon to switch to the light pencil (for intermediate measurements). Drag the outer handle of the compass up the circle to draw an arc. Then rotate the compass arm down, and draw another arc.

6. Move the compass to point B and repeat.

7. Choosing the Line tool will hide the compass (any tool will do, but we need the Line tool), showing your arcs:

8. Draw a line between the intersections of the pairs of arcs. This is the perpendicular bisector, and its intersection with the original line is the midpoint of line AB.

9. To make the bisector longer, tap and hold on it, tap again to show the menu, then tap Info in the menu. Turn on Extends. Now pick any two points on the line, connect them, and delete the original line.

# Problem of the Day: 1/29/13

In the rhombus below, angles A and B are 120°, and C and D are 60°. If AC = 8 cm, what is the radius of the circle? Express your answer with simplified radicals.

Solution to yesterday’s problem:

To find the greatest common factor of two numbers, find the prime factorization of the numbers and multiply the factors that occur in both numbers. (This method contrasts with the least common multiple, in which we multiply all the factors except the ones that occur in both numbers. We multiply more numbers to get the LCM, the multiple.) Luckily for us, we already have the prime factorization!

The numbers are $2\times2\times2\times3\times5$ and $2\times2\times3\times3\times5\times5$. Notice that both numbers have at least 2 twos, 1 three, and 1 five. Multiplying these factors together, we have $2\times2\times3\times5$, which yields 60.

# Isosceles Tutorial: Bisecting an Angle Traditionally

Bisecting an angle is a common task in geometry. There is a shortcut in Isosceles to construct a bisector, but sometimes one may need to do this traditionally. First, make sure you have Shows Intersections turned on in the Canvas pane of the Settings.

1. Tap to create a point.

2. Select the Line tool and drag from the point to create one leg of the angle.

3. Draw the other leg. Your angle should look somewhat like this:

4. Select the Compass tool. The compass will appear in blue across the diagram.

5. Drag the center handle of the compass to point A. Resize the compass if necessary by dragging the outer handle toward the center.

6. Tap the pencil icon to change to the light gray pencil. Drag the outer handle across the angle to draw an arc. Be sure to keep your finger close to the blue circle so that the compass does not resize.

7. Move the compass to one of the intersections between the arc and the angle. Draw another arc in the center of the angle. Repeat this step with the other intersection. Then select the Line tool to hide the compass. Your diagram should look similar to this:

8. Draw a line from point A that passes reasonably close to the intersection of the last two arcs. Isosceles will snap the line to pass through the intersection. This new line, AD, is the angle bisector of angle BAC.

# My Grapher Tutorial: Tangent Lines

You can use My Grapher to calculate a line tangent to a curve at a given point. Note: In this tutorial, “iPhone” refers to either an iPhone or an iPod touch.

1. iPad: Tap the + button and type in an equation for a curve. iPhone: Tap the list icon, then tap the + button to type in an equation for a curve.

2. If you graphed the equation shown above, you should get a graph like this:

3. iPad: Tap the gear icon, then tap Values. Choose Calculus from the two tabs at the top. iPhone: Tap and hold the graph, then tap Function Tool. Choose Calculus from the two tabs at the bottom. The options should look like this:

4. Tap X Value to choose which point the line should be tangent to. On iPhone, type the number in with the number keyboard.

5. Tap Done, and an equation should appear below the X Value option. Tap the equation to graph it.