$$ riangle A B C $$ is an isosceles triangle with $$ A C=B C $$ and $$ A B=144 \mathrm{~cm} $$. It is given that the perimeter of $$ = riangle A B C $$ is 384 cm. Find the radius of the circumscribed circle of $$ riangle A B C $$.

Question

$$ 	riangle A B C $$ is an isosceles triangle with $$ A C=B C $$ and $$ A B=144 \mathrm{~cm} $$. It is given that the perimeter of $$ =	riangle A B C $$ is 384 cm. Find the radius of the circumscribed circle of $$ 	riangle A B C $$.

UpStudy AI · Accepted Answer

1. 由題意得知$$ \triangle ABC $$為等腰三角形，且$$ AC=BC $$，邊長分別為  
   $$
   AB = 144\text{ cm},\quad AC = BC = x
   $$
   三角形的周長為384 cm，因此：
   $$
   144 + 2x = 384
   $$
   解得：
   $$
   2x = 384 - 144 = 240\quad\Rightarrow\quad x = 120\text{ cm}
   $$

2. 選取邊$$ AB $$為底邊，設底邊中點為$$ D $$，則：
   $$
   AD = DB = \frac{144}{2} = 72\text{ cm}
   $$
   設從頂點$$ C $$做到底邊$$ AB $$的垂線，垂足為$$ D $$，故可利用畢氏定理求出高度$$ h $$：
   $$
   h^2 + 72^2 = 120^2
   $$
   整理得：
   $$
   h^2 = 120^2 - 72^2 = 14400 - 5184 = 9216
   $$
   因此：
   $$
   h = \sqrt{9216} = 96\text{ cm}
   $$

3. 利用底邊與高度計算面積$$ \Delta $$：
   $$
   \Delta = \frac{1}{2} \times 144 \times 96 = 6912\text{ cm}^2
   $$

4. 利用三角形的邊長與面積求外接圓半徑$$ R $$：
   $$
   R = \frac{abc}{4\Delta}
   $$
   其中三邊分別為：
   $$
   a = 144\text{ cm},\quad b = 120\text{ cm},\quad c = 120\text{ cm}
   $$
   代入得：
   $$
   R = \frac{144 \times 120 \times 120}{4 \times 6912}
   $$
   先計算分子：
   $$
   144 \times 120 \times 120 = 144 \times 14400 = 2073600
   $$
   分母：
   $$
   4 \times 6912 = 27648
   $$
   因此：
   $$
   R = \frac{2073600}{27648}
   $$

5. 化簡計算：
   $$
   \frac{2073600}{27648} = \frac{2073600 \div 144}{27648 \div 144} = \frac{14400}{192} = \frac{14400 \div 16}{192 \div 16} = \frac{900}{12} = 75
   $$
   故外接圓半徑為：
   $$
   R = 75\text{ cm}
   $$
等腰三角形 $$ \triangle ABC $$ 的底邊 $$ AB = 144\text{ cm} $$，兩腰 $$ AC = BC = 120\text{ cm} $$。利用畢氏定理計算高 $$ h = 96\text{ cm} $$，面積 $$ \Delta = 6912\text{ cm}^2 $$。根據外接圓半徑公式 $$ R = \frac{abc}{4\Delta} $$，計算得 $$ R = 75\text{ cm} $$。

is an isosceles triangle with and . It is given that the perimeter of
is 384 cm. Find the radius of the circumscribed circle of .

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is an isosceles triangle with and . It is given that the perimeter of is 384 cm. Find the radius of the circumscribed circle of .