Question

sketch the figure described. explain how to construct it. then do the construction.
\\(\overleftrightarrow{xy} \perp \overleftrightarrow{yz}\\)

explain how to construct the figure accurately. what is the first step?
a. plot the points x, y, and z
b. draw \\(\overleftrightarrow{xy}\\) perpendicular to \\(\overleftrightarrow{yz}\\).
c. draw \\(\overleftrightarrow{xy}\\).
d. draw \\(\overleftrightarrow{xz}\\).

what is the second step of the construction?
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 \\(\overleftrightarrow{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 \\(\overleftrightarrow{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 \\(\overleftrightarrow{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 \\(\overleftrightarrow{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.

Explanation:

Identify the correct figure representing \(\overleftrightarrow{XY} \perp \overleftrightarrow{YZ}\)

The notation \(\overleftrightarrow{XY} \perp \overleftrightarrow{YZ}\) describes two perpendicular lines intersecting at point \(Y\).

• Figure A shows line \(\overleftrightarrow{XY}\) perpendicular to line \(\overleftrightarrow{YZ}\) intersecting at \(Y\).
• Figure B shows a perpendicular line that does not pass through \(Y\).

Determine the first step of the construction

To construct the perpendicular lines:

• First, establish the base line by drawing line \(\overleftrightarrow{XY}\).
• Therefore, the first step is to draw \(\overleftrightarrow{XY}\).

Determine the second step of the construction

To construct a perpendicular line through point \(Y\) on line \(\overleftrightarrow{XY}\):

• Set the compass to a convenient length.
• Place the compass point on \(Y\) and draw an arc intersecting \(\overleftrightarrow{XY}\) on the side of \(Y\) opposite \(X\). Label this point \(A\).
• This creates segment \(\overline{XA}\) with \(Y\) as its midpoint.

Determine the final step of the construction

With \(Y\) as the midpoint of segment \(\overline{XA}\):

• Constructing the perpendicular bisector of \(\overline{XA}\) will yield a line perpendicular to \(\overleftrightarrow{XY}\) passing through \(Y\).
• Label any point on this perpendicular bisector (other than \(Y\)) as \(Z\) to define line \(\overleftrightarrow{YZ}\).

Answer:

Question 1

• (A) Figure showing line XY perpendicular to line YZ intersecting at Y (Correct answer)
• (B) Figure showing a perpendicular line not passing through Y

Question 2

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

Question 3

• (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 \(\overleftrightarrow{XY}\). 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 \(\overleftrightarrow{XY}\). 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 \(\overleftrightarrow{XZ}\). 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 4

• (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.