# Oppgaver: Trekantgeometri :::::::::::::::{exercise} Oppgave 1 --- level: 1 --- ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a :::{plot} width: 100% align: right axis: off axis: equal triangle: svs=(3, 45, 5), angle-labels=(A=numeric, B=numeric) fontsize: 30 ::: En trekant $\triangle ABC$ er vist i figuren til høyre. Bestem $\angle C$. :::{clear} ::: ::::{admonition} Fasit --- class: answer, dropdown --- $$ \angle C = 36.39 \degree. $$ :::: ::::::::::::: :::::::::::::{tab-item} b :::{plot} width: 100% align: right axis: off axis: equal triangle: svs=(3, 90, 3), angle-labels=(C=numeric) fontsize: 30 ::: Figuren til høyre vises en trekant $\triangle ABC$. Bestem $\angle B$. :::{clear} ::: ::::{admonition} Fasit --- class: answer, dropdown --- $$ \angle B = 45 \degree. $$ :::: ::::::::::::: :::::::::::::{tab-item} c :::{plot} width: 100% align: right axis: off axis: equal triangle: svs=(2, 110, 2), angle-labels=(A=numeric, B=numeric) fontsize: 30 ::: En trekant $\triangle ABC$ er vist i figuren til høyre. Bestem $\angle C$. :::{clear} ::: ::::{admonition} Fasit --- class: answer, dropdown --- $$ \angle C = 35 \degree. $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 2 --- level: 1 --- ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a :::{plot} width: 100% align: right axis: off axis: equal triangle: sss=(3, 2*sqrt(3), sqrt(3)), angle-labels=(B=numeric), angles=(A, B), side-labels=(AB=exact, BC=exact) fontsize: 30 ::: I figuren til høyre vises en trekant $\triangle ABC$. Bestem $CA$. :::{clear} ::: ::::{answer} $$ CA = \sqrt{3}. $$ :::: ::::::::::::: :::::::::::::{tab-item} b :::{plot} width: 100% align: right axis: off axis: equal triangle: sss=(5, 10, 5*sqrt(3)), angle-labels=(B=numeric), angles=(A, B), side-labels=(AB=exact, BC=exact), angle-radius=40 fontsize: 34 ::: En trekant $\triangle ABC$ er vist i figuren til høyre. Bestem $AC$. :::{clear} ::: ::::{answer} $$ AC = 5 \sqrt{3}. $$ :::: ::::::::::::: :::::::::::::{tab-item} c :::{plot} width: 100% align: right axis: off axis: equal triangle: sss=(2 * sqrt(3), 2, 4), angle-labels=(A=numeric), angles=(A, B), side-labels=(BC=exact), angle-radius=25 fontsize: 30 ::: En trekant $\triangle ABC$ er vist i figuren til høyre. Bestem $AB$ og $AC$. :::{clear} ::: ::::{answer} $$ AB = 2 \sqrt{3} \and AC = 4. $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 3 ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a :::{plot} width: 100% align: right axis: off axis: equal triangle: sss=(4, 5, 3), angles=(A, B, C), side-labels=(BC=exact, AB=exact), angle-radius=30 fontsize: 30 ::: En trekant $\triangle ABC$ er vist i figuren til høyre. Bestem $CA$. :::{clear} ::: ::::{answer} $$ CA = 3. $$ :::: ::::::::::::: :::::::::::::{tab-item} b :::{plot} width: 100% align: right axis: off axis: equal triangle: svs=(3, 90, 2), angles=(A, B, C), side-labels=(BC=exact, CA=exact), angle-radius=25 fontsize: 30 ::: En trekant $\triangle ABC$ er vist i figuren til høyre. Bestem $AB$. :::{clear} ::: ::::{answer} $$ AB = 3. $$ :::: ::::::::::::: :::::::::::::{tab-item} c :::{plot} width: 100% align: right axis: off axis: equal triangle: sss=(6 * cos(pi/3), 6 * sin(pi/3), 6), angles=(A, B, C), side-labels=(BC=exact, AB=exact), angle-radius=30, label-offset=20 fontsize: 30 ::: En trekant $\triangle ABC$ er vist i figuren til høyre. Bestem $AC$. :::{clear} ::: ::::{answer} $$ AC = 6. $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 4 ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a :::{plot} width: 100% align: right axis: off axis: equal triangle: sss=(4, 4, 4), angles=(A, B), side-labels=(AB=exact), angle-radius=30, angle-labels=(A=numeric, B=numeric) fontsize: 30 vline: 2, 0, 2*sqrt(3), dashed, gray let: ds = 0.3 line-segment: (2 + ds, 0), (2 + ds, ds), solid, gray line-segment: (2 + ds, ds), (2, ds), solid, gray text: 2, 0.5 * 2 * sqrt(3), "$h$", center-right ::: En trekant $\triangle ABC$ er vist i figuren til høyre. I trekanten er $AB = 4$. Bestem høyden $h$ i trekanten. :::{clear} ::: ::::{answer} $$ h = 2 \sqrt{3}. $$ :::: ::::::::::::: :::::::::::::{tab-item} b :::{plot} width: 100% align: right axis: off axis: equal triangle: sss=(4, 4, 4), angles=(A, B), angle-radius=30, angle-labels=(A=numeric, B=numeric) fontsize: 30 vline: 2, 0, 2*sqrt(3), dashed, gray let: ds = 0.3 line-segment: (2 + ds, 0), (2 + ds, ds), solid, gray line-segment: (2 + ds, ds), (2, ds), solid, gray text: 2, 0.5 * 2 * sqrt(3), "$3$", center-right ::: En trekant $\triangle ABC$ er vist i figuren til høyre. Bestem omkretsen til trekanten. :::{clear} ::: ::::{answer} $$ 6 \sqrt{3} $$ :::: ::::::::::::: :::::::::::::{tab-item} c :::{plot} width: 100% align: right axis: off axis: equal triangle: sss=(4, 4, 4), angles=(A, B), angle-radius=30, side-text=(AB="$g$"), angle-labels=(A=numeric, B=numeric) fontsize: 30 vline: 2, 0, 2*sqrt(3), dashed, gray let: ds = 0.3 line-segment: (2 + ds, 0), (2 + ds, ds), solid, gray line-segment: (2 + ds, ds), (2, ds), solid, gray text: 2, 0.5 * 2 * sqrt(3), "$h$", center-right ::: Arealet $T$ av en trekant er gitt ved $$ T = \dfrac{1}{2} \cdot g \cdot h. $$ Trekanten til høyre er en likesidet trekant som har arealet $T = 3$. Bestem omkretsen til trekanten. :::{clear} ::: ::::{answer} $$ 6 \sqrt[4]{3} $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 5 I en sirkel med radius $4$ er det tegnet inn en trekant $\triangle ABC$ der $A$ er sentrum i sirkelen, og $B$ og $C$ ligger på sirkelperiferien. :::{plot} width: 100% align: right axis: off axis: equal let: r = 4 circle: (0, 0), r, red triangle: points=((0, 0), (r, 0), (r * cos(120 * pi/180), r * sin(120 * pi/180))), angles=(A), angle-radius=30, angle-labels=(A=numeric), angle-radius=25, angle-text-offset=14 fontsize: 30 nocache: ::: :::{clear} ::: ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a Bestem høyden $h$ til trekanten relativ til grunnlinja $AB$. ::::{answer} $$ h = 2 \sqrt{3}. $$ :::: ::::::::::::: :::::::::::::{tab-item} b Bestem arealet av trekanten. ::::{answer} $$ 4 \sqrt{3} $$ :::: ::::::::::::: :::::::::::::{tab-item} c Bestem $BC$. ::::{answer} $$ BC = 4 \sqrt{3}. $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 6 :::{plot} fontsize: 30 ticks: off align: right axis: equal width: 100% let: r = 1 let: v = 120 circle: (0, 0), r, red point: (-r * cos(v * pi/180), -r * sin(v * pi/180)) point: (0, 0) point: (-r, 0) line-segment: (0, 0), (-r, 0), solid, blue line-segment: (0, 0), (-r * cos(v * pi/180), -r * sin(v * pi/180)), solid, blue line-segment: (-r, 0), (-r * cos(v * pi/180), -r * sin(v * pi/180)), solid, blue angle-arc: (-r, 0), 0.4, 0, -30 let: u = -30 text: -r + 0.5 * cos(u * pi/180), 0.3 * sin(u * pi/180), "$30^\circ$", center-right text: -r * cos(v * pi/180), -r * sin(v * pi/180), "$P$", bottom-right ::: I et koordinatsystem er det tegnet inn en trekant og sirkel med radius $6$. To av hjørnene til trekanten ligger på sirkelen, og det tredje hjørnet er i sentrum av sirkelen. Bestem koordinatene til punktet $P$ på figuren. :::{clear} ::: ::::{answer} $$ P\left(3, -3\sqrt{3}\right) $$ :::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 7 :::{plot} fontsize: 30 width: 100% align: right axis: off axis: equal triangle: sss=(4, 2*sqrt(3), 2), angles=(B, C), angle-labels=(B=numeric), angle-radius=20, angle-label-offset=20, side-labels=(BC=exact) triangle: sss=(2, sqrt(3), 1), angles=(C), corner-labels=none, angle-radius=20, side-labels=(CA=exact) text: cos(pi/3), sin(pi/3), "$E$", top-left text: 2, 0, "$D$", bottom-center ::: I figuren til høyre er * $AE = 1$ * $BC = 2\sqrt{3}$. La $T_{\triangle ABC}$ være arealet til trekanten $\triangle ABC$ og $T_{\triangle ADE}$ være arealet til trekanten $\triangle ADE$. Bestem forholdet $$ \dfrac{T_{\triangle ABC}}{T_{\triangle ADE}} $$ :::{clear} ::: ::::{answer} $$ \dfrac{T_{\triangle ABC}}{T_{\triangle ADE}} = 4. $$ :::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 8 I figuren nedenfor vises en trekant. Bestem $x$, $y$ og $z$. :::{plot} width: 60% fontsize: 30 axis: off axis: equal triangle: sss=(10, 5*sqrt(3), 5), angles=(B, C), angle-labels=(B=numeric), angle-radius=50, angle-text-offset=20, side-text=(BC="$y$", CA="$x$"), label-offset=20 vline: 5 * cos(pi/3), 0, 5 * sin(pi/3), dashed, gray let: ds = 0.7 line-segment: (5 * cos(pi/3) + ds, 0), (5 * cos(pi/3) + ds, ds), solid, gray line-segment: (5 * cos(pi/3) + ds, ds), (5 * cos(pi/3), ds), solid, gray text: 5 * cos(pi / 3), 0, "$D$", bottom-center bar: (0, -1), 10, h text: 5, -1, "$10$", bottom-center text: 5 * cos(pi/3), 0.5 * 5 * sin(pi/3), "$z$", center-right ::: ::::{answer} $$ x = 5 \and y = 5 \sqrt{3} \and z = \dfrac{5\sqrt{3}}{2} $$ :::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 9 Arealet av figuren nedenfor er $12 \sqrt{3}$. Bestem $CM$. :::{plot} width: 60% fontsize: 30 axis: off axis: equal let: Ax = 0 let: Ay = 0 let: Bx = 6 let: By = 0 let: Cx = 3 let: Cy = 3 * sqrt(3) let: Mx = (Ax + Bx + Cx) / 3 let: My = (Ay + By + Cy) / 3 triangle: points=((Ax, Ay), (Mx, My), (Cx, Cy)), angles=(A, C), angle-radius=40, corner-labels=none, angle-labels=(A=numeric, C=numeric), angle-text-offset=20 triangle: points=((Mx, My), (Bx, By), (Cx, Cy)), angles=(B, C), angle-radius=40, corner-labels=none, angle-labels=(B=numeric, C=numeric), angle-text-offset=20 text: Ax, Ay, "$A$", bottom-left text: Bx, By, "$B$", bottom-right text: Cx, Cy, "$C$", top-center text: Mx, My - 0.2, "$M$", bottom-center ::: ::::{answer} $$ CM = 2 \sqrt{6} $$ :::: :::::::::::::::