Math / Geometry

Question(1) For each of the following descriptions, rewrite the statement using geometric notation and draw each figure. (a) Point $B$ on ray $W Q$ . (b) Line $R N$ bisects line segment $Q T$ . (C) Line $F G$ is parallel to line $T R$ . (2) Point $M$ is on $\overline{Q R} . Q R=20 \mathrm{~cm}$ . If $M R=12 \mathrm{~cm}$ , then what is $Q M$ ? (3) Draw a counterexample for each conditional statement. (a) If $\overline{F G} \cong \overline{G H}$ , then point $G$ is on $\overline{F H}$ . (b) If $\overleftrightarrow{A B}$ and $\overline{C D}$ are not parallel, then they must intersect.

Studdy Solution

STEP 1

1. We need to understand the geometric terms used in the description to translate them into geometric notation.
2. We should be able to draw the described figures based on the geometric notation.
3. For the second problem, we need to understand the properties of line segments and use basic arithmetic to find the missing length.
4. For the third problem, we need to understand the concept of counterexamples in the context of geometric statements.

STEP 2

1. Translate each description into geometric notation and draw the corresponding figures.
2. Calculate the length of $\overline{QM}$ given the lengths $QR$ and $MR$ .
3. Provide counterexamples for the given conditional statements.

STEP 3

(a) Translate "Point $B$ on ray $W Q$ " into geometric notation.
$B \in \overrightarrow{WQ}$

STEP 4

(a) Draw the figure for point $B$ on ray $\overrightarrow{WQ}$ .
\begin{array}{c} \begin{tikzpicture} \draw[->] (0,0) -- (4,0); \node at (0,-0.3) {W}; \node at (4,-0.3) {Q}; \node at (2,0.3) {B}; \filldraw (2,0) circle (2pt); \end{tikzpicture} \end{array}

STEP 5

(b) Translate "Line $R N$ bisects line segment $Q T$ " into geometric notation.
$RN \text{ bisects } \overline{QT}$

STEP 6

(b) Draw the figure for line $RN$ bisecting line segment $\overline{QT}$ .
\begin{array}{c} \begin{tikzpicture} \draw (0,0) -- (4,0); \draw[dashed] (2,-1) -- (2,1); \node at (0,-0.3) {Q}; \node at (4,-0.3) {T}; \node at (2,-1.3) {R}; \node at (2,1.3) {N}; \filldraw (2,0) circle (2pt); \end{tikzpicture} \end{array}

STEP 7

(c) Translate "Line $F G$ is parallel to line $T R$ " into geometric notation.
$FG \parallel TR$

STEP 8

(c) Draw the figure for line $FG$ parallel to line $TR$ .
\begin{array}{c} \begin{tikzpicture} \draw (0,0) -- (4,0); \draw (0,1) -- (4,1); \node at (0,-0.3) {T}; \node at (4,-0.3) {R}; \node at (0,1.3) {F}; \node at (4,1.3) {G}; \end{tikzpicture} \end{array}

STEP 9

Given $QR = 20 \, \mathrm{cm}$ and $MR = 12 \, \mathrm{cm}$ , find $QM$ using the segment addition property.
$QR = QM + MR$

STEP 10

Substitute the given lengths into the equation and solve for $QM$ .
$20 \, \mathrm{cm} = QM + 12 \, \mathrm{cm}$ $QM = 20 \, \mathrm{cm} - 12 \, \mathrm{cm}$ $QM = 8 \, \mathrm{cm}$

STEP 11

(a) Provide a counterexample for the statement: "If $\overline{FG} \cong \overline{GH}$ , then point $G$ is on $\overline{FH}$ ."
Consider points $F$ , $G$ , and $H$ such that $F$ , $G$ , and $H$ are not collinear. Draw the counterexample where $\overline{FG} \cong \overline{GH}$ but $G$ is not on $\overline{FH}$ .
\begin{array}{c} \begin{tikzpicture} \draw (0,0) -- (2,2); \draw (2,2) -- (4,0); \node at (0,-0.3) {F}; \node at (2,2.3) {G}; \node at (4,-0.3) {H}; \filldraw (0,0) circle (2pt); \filldraw (2,2) circle (2pt); \filldraw (4,0) circle (2pt); \end{tikzpicture} \end{array}

STEP 12

(b) Provide a counterexample for the statement: "If $\overleftrightarrow{AB}$ and $\overline{CD}$ are not parallel, then they must intersect."
Consider $\overleftrightarrow{AB}$ and $\overline{CD}$ as skew lines in 3D space. Draw the counterexample where the lines are not parallel and do not intersect.
\begin{array}{c} \begin{tikzpicture} \draw (0,0) -- (4,0); \draw (1,-1) -- (1,3); \node at (0,-0.3) {A}; \node at (4,-0.3) {B}; \node at (1,-1.3) {C}; \node at (1,3.3) {D}; \end{tikzpicture} \end{array}
Solution:
1. (a) Point $B$ on ray $\overrightarrow{WQ}$ : $B \in \overrightarrow{WQ}$ (b) Line $RN$ bisects line segment $\overline{QT}$ : $RN \text{ bisects } \overline{QT}$ (c) Line $FG$ is parallel to line $TR$ : $FG \parallel TR$
2. $QM = 8 \, \mathrm{cm}$
3. (a) Counterexample for $\overline{FG} \cong \overline{GH}$ with $G$ not on $\overline{FH}$ . (b) Counterexample for $\overleftrightarrow{AB}$ and $\overline{CD}$ not parallel and not intersecting.

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Studdy Solution

STEP 1

STEP 2

1. Translate each description into geometric notation and draw the corresponding figures.
2. Calculate the length of $\overline{QM}$ given the lengths $QR$ and $MR$ .
3. Provide counterexamples for the given conditional statements.

STEP 3

(a) Translate "Point $B$ on ray $W Q$ " into geometric notation.
$B \in \overrightarrow{WQ}$

STEP 4

(a) Draw the figure for point $B$ on ray $\overrightarrow{WQ}$ .
\begin{array}{c} \begin{tikzpicture} \draw[->] (0,0) -- (4,0); \node at (0,-0.3) {W}; \node at (4,-0.3) {Q}; \node at (2,0.3) {B}; \filldraw (2,0) circle (2pt); \end{tikzpicture} \end{array}

STEP 5

(b) Translate "Line $R N$ bisects line segment $Q T$ " into geometric notation.
$RN \text{ bisects } \overline{QT}$

STEP 6

STEP 7

(c) Translate "Line $F G$ is parallel to line $T R$ " into geometric notation.
$FG \parallel TR$

STEP 8

(c) Draw the figure for line $FG$ parallel to line $TR$ .
\begin{array}{c} \begin{tikzpicture} \draw (0,0) -- (4,0); \draw (0,1) -- (4,1); \node at (0,-0.3) {T}; \node at (4,-0.3) {R}; \node at (0,1.3) {F}; \node at (4,1.3) {G}; \end{tikzpicture} \end{array}

STEP 9

Given $QR = 20 \, \mathrm{cm}$ and $MR = 12 \, \mathrm{cm}$ , find $QM$ using the segment addition property.
$QR = QM + MR$

STEP 10

Substitute the given lengths into the equation and solve for $QM$ .
$20 \, \mathrm{cm} = QM + 12 \, \mathrm{cm}$ $QM = 20 \, \mathrm{cm} - 12 \, \mathrm{cm}$ $QM = 8 \, \mathrm{cm}$

STEP 11

STEP 12

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Studdy Solution

STEP 1

STEP 2

1. Translate each description into geometric notation and draw the corresponding figures.2. Calculate the length of QM‾\overline{QM}QM​ given the lengths QRQRQR and MRMRMR.3. Provide counterexamples for the given conditional statements.

STEP 3

(a) Translate "Point BBB on ray WQW QWQ" into geometric notation.B∈WQundefined B \in \overrightarrow{WQ} B∈WQ​

STEP 4

(a) Draw the figure for point BBB on ray WQundefined\overrightarrow{WQ}WQ​.\begin{array}{c} \begin{tikzpicture} \draw[->] (0,0) -- (4,0); \node at (0,-0.3) {W}; \node at (4,-0.3) {Q}; \node at (2,0.3) {B}; \filldraw (2,0) circle (2pt); \end{tikzpicture} \end{array}

STEP 5

(b) Translate "Line RNR NRN bisects line segment QTQ TQT" into geometric notation.RN bisects QT‾ RN \text{ bisects } \overline{QT} RN bisects QT​

STEP 6

STEP 7

(c) Translate "Line FGF GFG is parallel to line TRT RTR" into geometric notation.FG∥TR FG \parallel TR FG∥TR

STEP 8

(c) Draw the figure for line FGFGFG parallel to line TRTRTR.\begin{array}{c} \begin{tikzpicture} \draw (0,0) -- (4,0); \draw (0,1) -- (4,1); \node at (0,-0.3) {T}; \node at (4,-0.3) {R}; \node at (0,1.3) {F}; \node at (4,1.3) {G}; \end{tikzpicture} \end{array}

STEP 9

Given QR=20 cmQR = 20 \, \mathrm{cm}QR=20cm and MR=12 cmMR = 12 \, \mathrm{cm}MR=12cm, find QMQMQM using the segment addition property.QR=QM+MR QR = QM + MR QR=QM+MR

STEP 10

Substitute the given lengths into the equation and solve for QMQMQM.20 cm=QM+12 cm 20 \, \mathrm{cm} = QM + 12 \, \mathrm{cm} 20cm=QM+12cm QM=20 cm−12 cm QM = 20 \, \mathrm{cm} - 12 \, \mathrm{cm} QM=20cm−12cm QM=8 cm QM = 8 \, \mathrm{cm} QM=8cm

STEP 11

STEP 12

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1. Translate each description into geometric notation and draw the corresponding figures.
2. Calculate the length of $\overline{QM}$ given the lengths $QR$ and $MR$ .
3. Provide counterexamples for the given conditional statements.

(a) Translate "Point $B$ on ray $W Q$ " into geometric notation.
$B \in \overrightarrow{WQ}$

(a) Draw the figure for point $B$ on ray $\overrightarrow{WQ}$ .
\begin{array}{c} \begin{tikzpicture} \draw[->] (0,0) -- (4,0); \node at (0,-0.3) {W}; \node at (4,-0.3) {Q}; \node at (2,0.3) {B}; \filldraw (2,0) circle (2pt); \end{tikzpicture} \end{array}

(b) Translate "Line $R N$ bisects line segment $Q T$ " into geometric notation.
$RN \text{ bisects } \overline{QT}$

(c) Translate "Line $F G$ is parallel to line $T R$ " into geometric notation.
$FG \parallel TR$

(c) Draw the figure for line $FG$ parallel to line $TR$ .
\begin{array}{c} \begin{tikzpicture} \draw (0,0) -- (4,0); \draw (0,1) -- (4,1); \node at (0,-0.3) {T}; \node at (4,-0.3) {R}; \node at (0,1.3) {F}; \node at (4,1.3) {G}; \end{tikzpicture} \end{array}

Given $QR = 20 \, \mathrm{cm}$ and $MR = 12 \, \mathrm{cm}$ , find $QM$ using the segment addition property.
$QR = QM + MR$

Substitute the given lengths into the equation and solve for $QM$ .
$20 \, \mathrm{cm} = QM + 12 \, \mathrm{cm}$ $QM = 20 \, \mathrm{cm} - 12 \, \mathrm{cm}$ $QM = 8 \, \mathrm{cm}$