Chapter 11: Loci and Construction

Solve · page 209: measurement: (ii) mAB=6cm, mZA=150° and mZB=60° Given the measurements of ADEF : mDE = 4.8 cm, mEF = 4 cm and mZE = 45°, draw altitudes of ADEF and find orthocentre. Construct the following triangles and find whether there exists any ambiguous case. (1) ABCD ; mBC =5 cm, mB = 62° and mCD = 4.7 cm Gi) AKLM; mLM =6 cm, m“M= 42° and mLN =5 cm

(ii) $m\overline{AB}=6\text{ cm}$, $m\angle A=150^\circ$ and $m\angle B=60^\circ$ [$\text{مائش}$]

<svg viewBox="0 0 340 240" font-size="13"> <defs> <marker id="arrow" viewBox="0 0 10 10" refX="5" refY="5" markerWidth="6" markerHeight="6" orient="auto-start-reverse"> <path d="M 0 0 L 10 5 L 0 10 z" fill="currentColor"/> </marker> </defs> <!-- Triangle vertices: A(70, 180), B(250, 180), C is impossible since angle sum > 180 --> <line x1="70" y1="180" x2="250" y2="180" stroke="currentColor" stroke-width="1.5"/> <text x="160" y="195" fill="currentColor" text-anchor="middle">6 cm</text> <text x="65" y="185" fill="currentColor" text-anchor="end">A</text> <text x="255" y="185" fill="currentColor">B</text> <text x="150" y="100" fill="currentColor" text-anchor="middle">Impossible Triangle</text> </svg>

Scale: $1\text{ cm} = 1\text{ cm}$

1. Draw line segment $\overline{AB}$ of length $6\text{ cm}$.
2. At point $A$, draw an angle of $150^\circ$ using a protractor.
3. At point $B$, draw an angle of $60^\circ$ using a protractor, noticing that the sum of angles ($150^\circ + 60^\circ = 210^\circ$) exceeds $180^\circ$, so no triangle can be constructed.

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Given the measurements of $\Delta DEF : m\overline{DE} = 4.8\text{ cm}, m\overline{EF} = 4\text{ cm}$ and $m\angle E = 45^\circ$, draw altitudes of $\Delta DEF$ and find orthocentre. [$\text{مائش}$]

<svg viewBox="0 0 340 240" font-size="13"> <defs> <marker id="arrow2" viewBox="0 0 10 10" refX="5" refY="5" markerWidth="6" markerHeight="6" orient="auto-start-reverse"> <path d="M 0 0 L 10 5 L 0 10 z" fill="currentColor"/> </marker> </defs> <polygon points="80,180 230,180 150,80" fill="none" stroke="currentColor" stroke-width="1.5"/> <line x1="150" y1="80" x2="150" y2="180" stroke="currentColor" stroke-dasharray="4,4"/> <line x1="80" y1="180" x2="180" y2="125" stroke="currentColor" stroke-dasharray="4,4"/> <text x="75" y="190" fill="currentColor" text-anchor="end">D</text> <text x="235" y="190" fill="currentColor">E</text> <text x="150" y="72" fill="currentColor" text-anchor="middle">F</text> <text x="155" y="140" fill="currentColor">H</text> <text x="155" y="195" fill="currentColor" text-anchor="middle">4.8 cm</text> </svg>

Scale: $1\text{ cm} = 1\text{ cm}$

1. Draw line segment $\overline{DE} = 4.8\text{ cm}$.
2. At vertex $E$, construct an angle of $45^\circ$ and mark point $F$ at a distance of $4\text{ cm}$, then join $DF$ to complete $\Delta DEF$.
3. Draw perpendicular lines (altitudes) from each vertex to its opposite side, and locate their point of intersection called the orthocentre $H$.

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5. (i) $\Delta BCD ; m\overline{BC} = 5\text{ cm}, m\angle B = 62^\circ$ and $m\overline{CD} = 4.7\text{ cm}$ [$\text{مائش}$]

<svg viewBox="0 0 340 240" font-size="13"> <polygon points="70,180 230,180 190,90" fill="none" stroke="currentColor" stroke-width="1.5"/> <text x="65" y="185" fill="currentColor" text-anchor="end">B</text> <text x="235" y="185" fill="currentColor">C</text> <text x="195" y="82" fill="currentColor">D</text> <text x="150" y="195" fill="currentColor" text-anchor="middle">5 cm</text> <text x="215" y="135" fill="currentColor">4.7 cm</text> <text x="95" y="170" fill="currentColor">62°</text> </svg>

Scale: $1\text{ cm} = 1\text{ cm}$

1. Draw line segment $\overline{BC} = 5\text{ cm}$.
2. At point $B$, make an angle of $62^\circ$.
3. With center $C$ and radius $4.7\text{ cm}$, draw an arc to intersect the ray from $B$, labeling the intersection point as $D$ and noting if an ambiguous case exists.

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5. (ii) $\Delta KLM ; m\overline{LM} = 6\text{ cm}, m\angle M = 42^\circ$ and $m\overline{LN} = 5\text{ cm}$ [$\text{مائش}$]

<svg viewBox="0 0 340 240" font-size="13"> <polygon points="80,180 240,180 170,95" fill="none" stroke="currentColor" stroke-width="1.5"/> <text x="75" y="185" fill="currentColor" text-anchor="end">L</text> <text x="245" y="185" fill="currentColor">M</text> <text x="175" y="87" fill="currentColor">N</text> <text x="160" y="195" fill="currentColor" text-anchor="middle">6 cm</text> <text x="120" y="130" fill="currentColor">5 cm</text> <text x="220" y="165" fill="currentColor">42°</text> </svg>

Scale: $1\text{ cm} = 1\text{ cm}$

1. Draw line segment $\overline{LM} = 6\text{ cm}$.
2. At vertex $M$, construct an angle of $42^\circ$.
3. With center $L$ and radius $5\text{ cm}$, draw an arc cutting the terminal side of the angle to locate point $N$, and check for any ambiguous case.