Question

a. connect the points x, y, and z with lines \\(\overleftrightarrow{xy}\\), \\(\overleftrightarrow{yz}\\), and \\(\overleftrightarrow{xz}\\).
b. set the compass to the length of \\(\overline{xy}\\). place the compass point on the point x and draw an arc that intersects \\(\overleftrightarrow{xy}\\) on the side of x opposite y. label that point a.
c. set the compass to the length of \\(\overline{xy}\\). place the compass point on the point y and draw an arc that intersects \\(\overleftrightarrow{xy}\\) on the side of y opposite x. label that point a.
d. set the compass to the length of \\(\overline{xz}\\). place the compass point on the point z and draw an arc that intersects \\(\overleftrightarrow{xz}\\) on the side of z opposite x. label that point a.

what is the final step of the construction?

a. construct the perpendicular bisector of \\(\overline{xa}\\). label any point on the bisector (that is not y) as z.
b. construct the perpendicular bisector of \\(\overline{ya}\\). label any point on the bisector (that is not x) as z.
c. construct the perpendicular bisector of \\(\overline{xy}\\). label any point on the bisector (that is not a) as z.
d. construct the perpendicular bisector of \\(\overline{za}\\). label any point on the bisector (that is not y) as x.

what does the final construction look like? choose the correct answer below.

Explanation:

Analyze the first construction step

The selected option is C: "Set the compass to the length of \(\overline{XY}\). Place the compass point on the point \(Y\) and draw an arc that intersects \(\overleftrightarrow{XY}\) on the side of \(Y\) opposite \(X\). Label that point \(A\)." This step creates a point \(A\) such that \(Y\) is the midpoint of segment \(\overline{XA}\), meaning \(XY = YA\).

Determine the final step of the construction

To construct a perpendicular line through point \(Y\), we need to construct the perpendicular bisector of the segment \(\overline{XA}\). Since \(Y\) is the midpoint of \(\overline{XA}\), the perpendicular bisector of \(\overline{XA}\) will pass directly through \(Y\). Thus, the final step is to construct the perpendicular bisector of \(\overline{XA}\) and label a point on it as \(Z\). This corresponds to option A.

Identify the correct final construction diagram

The construction involves:

1. A line containing points \(X\), \(Y\), and \(A\) in that order, with \(XY = YA\).
2. An arc centered at \(X\) and an arc centered at \(A\) with equal radii greater than \(XY\) intersecting above and below the line to construct the perpendicular bisector of \(\overline{XA}\).
3. The perpendicular line passing through \(Y\) with point \(Z\) labeled on it.

Diagram A correctly shows the arcs centered at \(X\) and \(A\) intersecting to form the perpendicular bisector through \(Y\).

Answer:

Question 1

• A. Connect the points X, Y, and Z with lines \(\overleftrightarrow{XY}\), \(\overleftrightarrow{YZ}\), and \(\overleftrightarrow{XZ}\).
• B. Set the compass to the length of \(\overline{XY}\). Place the compass point on the point X and draw an arc that intersects \(\overleftrightarrow{XY}\) on the side of X opposite Y. Label that point A.
• C. Set the compass to the length of \(\overline{XY}\). Place the compass point on the point Y and draw an arc that intersects \(\overleftrightarrow{XY}\) on the side of Y opposite X. Label that point A. (Correct answer)
• D. Set the compass to the length of \(\overline{XZ}\). Place the compass point on the point Z and draw an arc that intersects \(\overleftrightarrow{XZ}\) on the side of Z opposite X. Label that point A.

Question 2

• A. Construct the perpendicular bisector of \(\overline{XA}\). Label any point on the bisector (that is not Y) as Z. (Correct answer)
• B. Construct the perpendicular bisector of \(\overline{YA}\). Label any point on the bisector (that is not X) as Z.
• C. Construct the perpendicular bisector of \(\overline{XY}\). Label any point on the bisector (that is not A) as Z.
• D. Construct the perpendicular bisector of \(\overline{ZA}\). Label any point on the bisector (that is not Y) as X.

Question 3

• A. Diagram showing arcs centered at X and A intersecting to construct the perpendicular bisector of XA through Y. (Correct answer)
• B. Diagram showing incorrect arc placements.
• C. Diagram showing arcs centered at X and Y.
• D. Diagram showing a single large arc centered at Y.