Question

1. this diagram is a straightedge and compass construction of a line perpendicular to line ab passing through point c. which segment has the same length as segment ea? a. ec b. ed c. be d. bd (from unit 1, lesson 5.) 5. this diagram is a straightedge and compass construction. which triangle is equilateral? explain how you know. (from unit 1, lesson 4.) 6. in the construction, a is the center of one circle, and b is the center of the other. name the segments in the diagram that have the same length as segment ab.

Explanation:

Question 4

Step1: Recall compass - construction properties

In a straight - edge and compass construction of a perpendicular line, the compass is used to create equal - length segments. Points are constructed using the compass's radius. Since the same compass setting is used to create arcs from the same center point in the construction, if we consider the construction of the perpendicular line from point $C$ to line $AB$, and assume that the compass is set to a certain radius to mark points $E$ and other related points, the segments drawn with the same compass setting have the same length. Here, segment $EA$ and segment $EC$ are both drawn with the same compass setting from the same center point (in the process of the perpendicular - line construction).

Step1: Recall the definition of an equilateral triangle

An equilateral triangle has all three sides of equal length. In a straight - edge and compass construction, when circles are drawn with the same radius from different center points and the intersection points are connected, we can find equilateral triangles. In the given diagram, if we consider the circles and their intersections, assume that the circles are drawn with the same radius. Let's say we have circles centered at different points. The segments connecting the intersection points of these circles are equal in length because they are all equal to the radius of the circles used in the construction. For example, if we consider $\triangle XYZ$ (assuming appropriate points $X$, $Y$, and $Z$ which are intersection points of circles with the same radius), since the lengths of $XY$, $YZ$, and $ZX$ are all equal to the radius of the circles used in the construction, $\triangle XYZ$ is equilateral.

Step1: Recall circle - radius properties

Since $A$ is the center of one circle and $B$ is the center of the other, and in a circle, all radii are equal. Segment $AB$ is a distance between the centers of two circles. In the construction, the segments $AC$, $AD$, $BC$, and $BD$ are all radii of the respective circles. The circles centered at $A$ and $B$ are used in the construction, and the segments connecting the centers to the intersection points of the circles are radii. So, $AC = AD=AB$ (radii of the circle centered at $A$) and $BC = BD = AB$ (radii of the circle centered at $B$).

Answer:

A. $EC$

Question 5