Question

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Answer 1

$1.quadsin B=sqrt{1-cos^2B}=sqrt{1-frac{144}{169}}=sqrt{frac{25}{169}}=frac5{13}\sin B=frac{AC}{BC}Rightarrow DC=frac{AC}{sin B}=15cdotfrac{13}5=39\ 2.quadsinalphacdotcosalpha=frac13quadtimes2\ 2sinalphacosalpha=frac23quad+1\ 2sinalphacosalpha+1=frac23+1\ 2sinalphacosalpha+sin^2alpha+cos^2alpha=frac53\ (sinalpha+cosalpha)^2=frac53=frac{15}9\ sinalpha+cosalpha=sqrt{frac{15}9}=frac{sqrt{15}}3$

$3.quad S_{ABCOD}=S_{ABCD}-S_{COD}\ AC=BD=sqrt{AB^2+AD^2}=sqrt{225+64}=sqrt{289}=17\CO=frac12AC=frac{17}2=8,5\S_{COD}=frac12cdot CO^2cdotsinwidehat{COD}=frac12cdot8,5^2cdotfrac12=frac14cdot72frac14=18frac1{16}\S_{ADBC}=15cdot8=120\S_{ABCOD}=120-18frac1{16}=101frac{15}{16}$