Goal: Given a line and a point on the line, construct a perpendicular line through the given point.[br][br][b][color=#ff0000][u]Step 1[/u]: Find two points on the line that are equidistant from the given point.[/color][/b][br]- Select "Circle with Center through Point" in the 4th dropdown window.[br]- Click on P to center the circle at P, then click anywhere [u]on the line[/u] to the [u]left[/u] of P. You now have a circle centered at P, through point A.[br]- Select the "Point" tool (first option in the the 2nd dropdown window). Click on where the circle intersects the line, on the [u]right[/u] side of P. [br][br][i][b]**Whenever you place a point at an intersection, double check it that it shows up as [u]black[/u]. [br]If the point shows up as [color=#1e84cc]blue[/color], then it is [u]not[/u] on the intersection.**[/b][br][/i][br]You should now have points A and B as the two intersection points.[br][color=#ff0000]These two points on the line are [i][u]equidistant[/u] [/i]from P, because they are [u][i]on the same circle[/i][/u] centered at P.[/color][br][b][color=#ff7700][u][br]Step 2[/u]: Create two circles with the same radius, one centered at A and one centered at B.[/color][/b][br]- Use the "Circle with Center through Point" tool to make a circle, centered at A, that is a little bit larger than the circle centered at P. You should notice a point on A labelled "C."[br]- Select the "Compass" tool in the 4th dropdown window. Click on A and then C - this "fixes" the radius of your new circle so that it equals AC. [br]- Now click on point B. [br][color=#ff7700]You should now have two [i][u]intersecting[/u] [/i]circles, one centered at B, and one centered at A, that have the [u][i]same radius[/i][/u].[br][/color][br][b][color=#6aa84f][u]Step 3[/u]: Locate the intersection of the two circles. [/color][/b][br]- Use the "POINT" tool to label the point above P where the two large circles intersect. It should show up as point D.[br][b][color=#0000ff][u]Step 4[/u]: Construct a line through points D and P. [/color][br][/b]- Use the "Line" tool (first tool in the 3rd dropdown window), then click on points D and P.[br][b][color=#9900ff][u]Step 5[/u]: "lighten" the arc marks.[/color][br][/b]- Click “Move” (the 1st window) to allow you to click and drag objects. [br]- Click on one of the circles. [br]- Notice a menu just below the upper right corner. It has a circle and a triangle over three gray lines. Click on this menu to expand the color/outline options.[br]- Click on the "solid line segment" box. A dropdown menu of dotted lines will show up. Select the fourth "dotted line" option to make the circle have a thin, dotted outline.[br]- Repeat this for all of your circles. [br][br][b]Now, the two [u]lines[/u] should stand out visually, but the circles are still visible as a way to show your work.[/b]

Given a line and a point NOT on the line, we can construct a parallel line through this point using the theorem that [b][i]two lines perpendicular to the same line are parallel[/i][/b].[br][br][b][color=#ff0000][u]Step 1[/u]: Construct a perpendicular line through the given point.[/color][/b][br][i][b]a) Find two points on the line that are equidistant from point P.[/b][/i][br]- Use the "Construct Circle with Center through Point" tool --> Click on P --> Click on the line so that you have a circle that intersects the line at point C AND another point.[br]- Use the "Point" tool to label the other intersection point (it should show up as point D).[br][i][b]b) Find a new point below the line that is equidistant from the two points you just found.[/b][/i][br]- Use the "Construct Circle with Center through Point" tool to create a circle centered at D. A new point E should show up on the circle.[br]- Select the "Compass" tool. Click on point D & E to fix the radius, then center the circle at C. [br]- Now, you should have two circles with the same radius, one centered at D and one centered at C.[br]- Use the "Point" tool to label the intersection of these two circles. It should show up as point G.[i] If they don't intersect, use the "Move" tool to click and drag point E until the circles are big enough to intersect.[br][/i][b][i]c) Construct a line through P and F. [/i][/b][br]- Use the "Line" tool, then click both points.[br][br][b][color=#ff7700]** [u]Pause[/u]: Color-code before the next part **[/color][/b][br]- Make the color of the perpendicular line you just constructed [b]red[/b].[br]- Make the circles have [b]dotted outlines[/b] so that the lines you constructed show up more clearly.[br]- Pro-tip: you can hold the "CTRL" button while you click in order to select multiple objects at once.[br][br]Confirm your work is correct so far: If you click and drag either of the blue endpoints of the original segment, the red line should remain perpendicular to it.[br][br][b][color=#0000ff][u]Step 2[/u]: Construct a new line that is perpendicular to the red line, through point P. [/color][br]We'll use the same process as in problem #1:[/b][br]- Construct a circle centered at P to find and label two points on the red line that are equidistant from P.[br]- Construct a circle centered at one of these new points. [br]- Use the "Compass" tool to copy a circle with the same-size radius, centered at the other new point.[br]- Place one last point on the intersection of these congruent circles, then construct a line through this point and point P.[br][br]The original line and the new line will be parallel, because they are both perpendicular to the same line![br]You can confirm this is correct by clicking and dragging any blue points on the screen.

A rectangle is a quadrilateral with four sets of perpendicular sides.[br]You can construct a rectangle by constructing perpendiculars, as you did in the previous two tasks.[br][br]In this task, you will build a rectangle with width DC and height XY. [br][br][b][u]Step 1[/u]: [/b][color=#ff0000]Construct a [/color][b][color=#ff0000]perpendicular line through D[/color][/b], using the process shown in task #1. [br]Make this line [b][color=#ff0000]red[/color][/b][color=#ff0000].[br][/color][i]Make your circles have a thin dotted outline, and drag the points to make the circles smaller, before you move on (just to clear up the screen).[br][/i][b][u][br]Step 2[/u]: [/b][color=#ff7700]Construct a [/color][b][color=#ff7700]perpendicular line through C[/color], [/b]using the process shown in task #1. [br]Make this line [b][color=#ff7700]orange[/color][/b][color=#ff7700]. [br][/color][i]Make your circles have a thin dotted outline, [i]and drag the points to make the circles smaller, [/i]before you move on (just to clear up the screen).[br][br][/i][b][u]Step 3[/u]: [/b][color=#9900ff]Locate and label [b]point A[/b] [/color]by using the "Compass" tool to copy the correct length (make a circle with [b]radius XY[/b], [b]centered at D[/b]).[br][br][b][u][color=#ff0000]To rename a point[/color]:[/u] right click[/b] the point --> click on [b]"rename."[/b][br][b][u][br]Step 4[/u]: [/b][color=#0000ff]Construct a[b] perpendicular line through A[/b],[/color] using the process shown in task #1. [br]Make this line [b][color=#0000ff]blue[/color].[br][/b][b][u][br]Step 5[/u]: [/b][color=#9900ff]Locate and label [/color][b][color=#9900ff]point B [/color][/b]as the intersection of the sides.[br][br]We know that ABCD is a rectangle, because all of its sides are perpendicular by construction.[br][br][b]Clean up your work[/b] by making your circles dotted + dragging the points so that the circles are not too big. [br][br][b]Check your work: [/b][list][*]Click and move around points D and C to check that the sides remain [b]perpendicular[/b].[/*][*]Click and move around X and Y to check that the [b]height [/b]of the rectangle stays equal to XY.[/*][/list]

In this task, you will build the same rectangle from the same given information. However, this will show you with an easier way to do it.[br][br]You can use the fact that [b][i]opposite sides of a rectangle are equal in length[/i][/b] to construct the rectangle, rather than constructing multiple different perpendiculars.[br][br][b][u]Step 1[/u]: [/b][color=#ff0000]Construct a [/color][b][color=#ff0000]perpendicular line through D[/color][/b], using the process shown in task #1. [br]Make this line [b][color=#ff0000]red[/color][/b][color=#ff0000].[br][/color][i]Make your circles have a thin dotted outline, and drag the points to make the circles smaller, before you move on (just to clear up the screen).[br][br][/i][b][u]Step 2[/u]: [/b][color=#9900ff]Locate and label ([u]rename[/u]) [b]point A[/b] [/color]by using the "Compass" tool to copy the correct length (make a circle with [b]radius XY[/b], [b]centered at D[/b]).[br][br][u][b]Step 3[/b][/u][b]: [/b]Use the "Compass" tool to create a [color=#0000ff][b]circle with radius CD, centered at A[/b][/color]. Every point on this circle represents a point that is "CD" units from point A.[br][br][b][u]Step 4[/u]: [/b]Use the "Compass tool to create a [color=#0000ff][b]circle with radius XY, centered at C[/b][/color]. Every point on this circle represents a point that is "XY" units from point C.[b][br][br][u]Step 5[/u]: [/b][color=#9900ff]Locate and label ([u]rename[/u]) [/color][b][color=#9900ff]point B [/color][/b]as the [b][color=#0000ff]intersection [/color][/b]of the circles. This is the only point that is exactly "XY units from point C [u][b]AND[/b][/u] "DC" units from point A.[br][br]This time, we know that ABCD is a rectangle, because its opposite sides have the same length, and it contains a perpendicular set of sides.[br][br][b]Clean up your work[/b] by making your circles dotted + dragging the points so that the circles are not too big. [br][br][b]Check your work: [/b][list][*]Click and move around points D and C to check that the sides remain [b]perpendicular[/b].[/*][*]Click and move around X and Y to check that the [b]height [/b]of the rectangle stays equal to XY.[/*][/list]

a) Use the "Segment" tool to [b]add the diagonals[/b] in rectangle ABCD below. [br]b) Place a point at the [b]intersection [/b]of these diagonals.[br][br]Move around points A, C, and D so the rectangle and its diagonals change. As you do so, investigate: What relationship(s) between the diagonals never seem to change, no matter the dimensions of the rectangle?

A square is a quadrilateral in which all four sides are congruent, and adjacent sides are perpendicular.[br][br]In other words, a square is an equilateral rectangle.[br][br]Using the skills from this assignment, construct a square from the given side length, shown below.