Question

lesson 9 practice problems

1. which construction can be used to determine whether point c is closer to point a or point b?

a. construct triangle abc.
b. construct a line perpendicular to segment ab through point c.
c. construct the bisector of angle acb.
d. construct the perpendicular bisector of segment ab.

1. the diagram is a straightedge and compass construction. lines l, m, and n are the perpendicular bisectors of the sides of triangle abc. select all the true statements.

a. point e is closer to point a than it is to point c.
b. point l is closer to point b than it is to point a.
c. point d is closer to point b than it is to point c.
d. point j is closer to point a than it is to point b or point c.
e point k is closer to point c than it is to point a or point b.
f. point l is closer to point c than it is to point a or point b.

1. decompose the figure into regions that are closest to each vertex. explain or show your reasoning.

Explanation:

Step1: Recall property of perpendicular bisector

The perpendicular bisector of a line - segment consists of all points that are equidistant from the endpoints of the segment. Points on one side of the perpendicular bisector of segment \(AB\) are closer to one endpoint, and points on the other side are closer to the other endpoint.

Step2: Answer question 1

To determine whether point \(C\) is closer to point \(A\) or point \(B\), we construct the perpendicular bisector of segment \(AB\). So the answer to question 1 is D.

Step3: Analyze properties of points and perpendicular bisectors for question 2

• For a point on the perpendicular bisector of a side of a triangle, it is equidistant from the two vertices of that side. For points not on the perpendicular bisector, we can use the fact that points on one - side of the perpendicular bisector of a segment are closer to one endpoint.
• Point \(E\) is on the perpendicular bisector of \(AC\), so it is equidistant from \(A\) and \(C\), so A is false.
• Point \(L\) is on the perpendicular bisector of \(AB\), so it is equidistant from \(A\) and \(B\), so B is false.
• Point \(D\) is on the perpendicular bisector of \(BC\), so it is equidistant from \(B\) and \(C\), so C is false.
• Point \(J\) is on the perpendicular bisector of \(AB\) and is on the side closer to \(A\) compared to \(B\) and is also closer to \(A\) compared to \(C\) (by observing the position relative to the perpendicular bisectors), so D is true.
• Point \(K\) is on the perpendicular bisector of \(AC\) and is on the side closer to \(C\) compared to \(A\) and also closer to \(C\) compared to \(B\) (by observing the position relative to the perpendicular bisectors), so E is true.
• Point \(L\) is on the perpendicular bisector of \(AB\), so it is equidistant from \(A\) and \(B\), so F is false.
• The correct answers for question 2 are D and E.

Step4: Answer question 3

Construct the perpendicular bisectors of the sides of the quadrilateral \(ABCD\). The regions are formed by the intersection of the half - planes determined by these perpendicular bisectors. Each region contains points that are closer to a particular vertex. For example, the region closest to vertex \(A\) is the intersection of the half - planes such that points in this region are closer to \(A\) than to \(B\), \(C\), or \(D\). We can construct the perpendicular bisectors of segments \(AB\), \(BC\), \(CD\), and \(DA\). The intersection of the half - planes on the side of each perpendicular bisector that contains the relevant vertex will give the regions closest to each vertex.

Answer:

1. D. Construct the perpendicular bisector of segment \(AB\).
2. D. Point \(J\) is closer to point \(A\) than it is to point \(B\) or point \(C\); E. Point \(K\) is closer to point \(C\) than it is to point \(A\) or point \(B\).
3. Construct the perpendicular bisectors of segments \(AB\), \(BC\), \(CD\), and \(DA\). The intersection of the half - planes on the side of each perpendicular bisector that contains the relevant vertex will give the regions closest to each vertex.