Problem 4-1
4-1 A
4-1 A Explanatioin
To make this construction I first made the segment AB. Then I used the Rotate around a point tool. This allowed me to choose the segment AB to rotate, point B as the center of the rotation, and then 60 degrees as the angle it was to be rotated, we we know that all the angles of an equilateral must be 60 degrees. This created segment A'B' where point B' is the same as point B. Then I made segment A'A to close the shape. I know that this makes an equilateral triangle because an equilateral triangle has all equal angles and sides.
4-1 B
4-1 B Explanation
First I made a circle with radius the segment AB. Then I created a perpendicular line to segment AB that went through point A. I made point C where the perpendicular line and the circle met. We know that AB is the same length as AC because they are both the radius of the circle. I then created a line parallel to AB that went through point C and a line perpendicular to AB that went through point B. These two knew lines met at point D. They are also perpendicular, which is why this angle also has 90 degrees. When moving any of the points, the angles do not change and the lengths of the segments all stay equal to one another.
4-1 C
4-1 C Explantion
First I made a circle with center A, and then points B and C on the circle. I created segments AB and AC, which have the same length because they are both radii of the circle. I then made segment BC to close the shape. Since AB and AC are the same length, the angles opposite of these sides (angle B and angle C) are also equal. We know this is isosceles triangle because it has two congruent sides and two congruent angles.
4-1 D
4-1 D Explanation
First I made circle with center A and radius AB. To make the diameter BB', I used the Reflect around Point to reflect point B around point A. This made me know for sure that segment BB' went through the center A and also stayed one straight line instead of two different segments. Then I created point C on the circle. No matter where point C moves around the circle, it stays 90 degrees because the angle is half the arc length between B and B'. Since this is the diameter, then the arc length is 180 degrees, so angle C will always stay 90 degrees.