Question

explain how to construct the figure accurately. what is the first step?

a. draw \\(\overleftrightarrow{xy}\\) perpendicular to \\(\overleftrightarrow{yz}\\).
b. draw \\(\overleftrightarrow{xz}\\).
c. draw \\(\overleftrightarrow{xy}\\).
d. plot the points x, y, and z

what is the second step of the construction?

a. 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.
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{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.
d. connect the points x, y, and z with lines \\(\overleftrightarrow{xy}\\), \\(\overleftrightarrow{yz}\\), and \\(\overleftrightarrow{xz}\\).

what is the final step of the construction?

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

Explanation:

Identify the target construction

$$ \overleftrightarrow{XY} \perp \overleftrightarrow{YZ} $$

Determine the first step

$$ \text{To construct a perpendicular line through point } Y \text{ on line } \overleftrightarrow{XY}, \text{ the base line must first be drawn: } \overleftrightarrow{XY}. $$

Determine the second step

$$ \text{To construct a perpendicular at } Y, \text{ we must find a point } A \text{ on } \overleftrightarrow{XY} \text{ such that } Y \text{ is the midpoint of segment } \overline{XA}. $$

$$ \text{Set the compass to the length of } \overline{XY}. \text{ Place the compass point on } Y \text{ and draw an arc that intersects } \overleftrightarrow{XY} \text{ on the side of } Y \text{ opposite } X. \text{ Label that point } A. $$

Determine the final step

$$ \text{Since } Y \text{ is the midpoint of } \overline{XA}, \text{ the perpendicular line through } Y \text{ is the perpendicular bisector of } \overline{XA}. $$

$$ \text{Construct the perpendicular bisector of } \overline{XA}. \text{ Label any point on the bisector (that is not } Y\text{) as } Z. $$

Answer:

Question 1

• A. Draw \(\overleftrightarrow{XY}\) perpendicular to \(\overleftrightarrow{YZ}\).
• B. Draw \(\overleftrightarrow{XZ}\).
• C. Draw \(\overleftrightarrow{XY}\). (Correct answer)
• D. Plot the points X, Y, and Z

Question 2

• A. 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)
• 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 \(\overline{XZ}\). 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.
• D. Connect the points X, Y, and Z with lines \(\overleftrightarrow{XY}\), \(\overleftrightarrow{YZ}\), and \(\overleftrightarrow{XZ}\).

Question 3

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