Question

g1.5 ws basic constructions
for exercises 1–9, draw a diagram similar to the given one. then do the
construction.

1. construct \\(\overline{qr}\\) congruent to \\(\overline{bc}\\). to start, draw
2. construct \\(\overline{lm}\\) so that \\(lm = 3bc\\).
3. construct \\(\overline{xy}\\) so that \\(xy = rs + tu\\).
4. construct \\(\overline{ab}\\) so that \\(ab = rs - tu\\).
5. construct \\(\angle j\\) so that \\(\angle j \cong \angle d\\)

Answer:

Problem 1: Construct $\overline{QR}$ congruent to $\overline{BC}$

Step 1: Draw a ray

Draw a ray with endpoint $Q$. This will be the starting point for constructing the congruent segment.

Step 2: Measure $\overline{BC}$

Using a compass, place the needle at $B$ and the pencil at $C$ to measure the length of $\overline{BC}$.

Step 3: Transfer the length

Without changing the compass width, place the needle at $Q$ and draw an arc that intersects the ray. Label the intersection point $R$. Now, $\overline{QR} \cong \overline{BC}$.

Problem 2: Construct $\overline{LM}$ so that $LM = 3BC$

Step 1: Draw a ray

Draw a ray with endpoint $L$.

Step 2: Measure $\overline{BC}$

Use a compass to measure the length of $\overline{BC}$ (needle at $B$, pencil at $C$).

Step 3: Mark the first segment

Place the compass needle at $L$ and draw an arc on the ray, marking point $A$ such that $LA = BC$.

Step 4: Mark the second segment

Keep the compass width, place the needle at $A$, and draw another arc, marking point $B$ such that $AB = BC$ (so $LB = 2BC$).

Step 5: Mark the third segment

Place the compass needle at $B$, draw an arc, marking point $M$ such that $BM = BC$. Now, $LM = LA + AB + BM = 3BC$.

Problem 3: Construct $\overline{XY}$ so that $XY = RS + TU$

Step 1: Draw a ray

Draw a ray with endpoint $X$.

Step 2: Measure $\overline{RS}$

Use a compass to measure the length of $\overline{RS}$ (needle at $R$, pencil at $S$).

Step 3: Mark $RS$ on the ray

Place the compass needle at $X$ and draw an arc on the ray, marking point $A$ such that $XA = RS$.

Step 4: Measure $\overline{TU}$

Use the compass to measure the length of $\overline{TU}$ (needle at $T$, pencil at $U$).

Step 5: Mark $TU$ from $A$

Keep the compass width, place the needle at $A$, and draw an arc on the ray, marking point $Y$ such that $AY = TU$. Now, $XY = XA + AY = RS + TU$.

Problem 4: Construct $\overline{AB}$ so that $AB = RS - TU$

Step 1: Draw a ray

Draw a ray with endpoint $A$.

Step 2: Measure $\overline{RS}$

Use a compass to measure the length of $\overline{RS}$ (needle at $R$, pencil at $S$).

Step 3: Mark $RS$ on the ray

Place the compass needle at $A$ and draw an arc on the ray, marking point $C$ such that $AC = RS$.

Step 4: Measure $\overline{TU}$

Use the compass to measure the length of $\overline{TU}$ (needle at $T$, pencil at $U$).

Step 5: Mark $TU$ from $C$ towards $A$

Place the compass needle at $C$ and draw an arc towards $A$ on the ray, marking point $B$ such that $CB = TU$. Now, $AB = AC - CB = RS - TU$.

Problem 5: Construct $\angle J$ so that $\angle J \cong \angle D$

Step 1: Draw a ray

Draw a ray with endpoint $J$ (this will be one side of $\angle J$).

Step 2: Measure $\angle D$

• Place the compass needle at vertex $D$. Draw an arc that intersects both sides of $\angle D$, marking points $E$ (on one side) and $F$ (on the other side).

Step 3: Transfer the arc to $\angle J$

• Without changing the compass width, place the needle at $J$ and draw an arc that intersects the ray from $J$, marking point $K$.

Step 4: Measure the distance between $E$ and $F$

• Use the compass to measure the distance between $E$ and $F$ (needle at $E$, pencil at $F$).

Step 5: Mark the intersection on the arc

• Keep the compass width, place the needle at $K$, and draw an arc that intersects the previously drawn arc at point $L$.

Step 6: Draw the second side of $\angle J$

• Draw a ray from $J$ through $L$. Now, $\angle J \cong \angle D$.

(Note: These are the step - by - step constructions for each of the given problems. The actual construction requires the use of a compass and straightedge with precise drawing.)