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Problem 536

Find, in radical form, the length of the radius of a circle circumscribed about an equilateral triangle, the length of whose side is 24 .

Expert verified

The length of the radius of the circle circumscribed about the equilateral triangle with a side length of 24 is 24.

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Chapter 28

In the figure, let \(\mathrm{ABCDE}\) be a regular pentagon touching the circle at points \(\mathrm{M}_{1}, \mathrm{M}_{2}, \mathrm{M}_{3}, \mathrm{M}_{4}\) and \(\mathrm{M}_{5}\). Assume that \(\mathrm{OA}=\mathrm{OB}=\mathrm{OC}=\mathrm{OD}=\mathrm{OE}\). (a) Prove that \(\triangle \mathrm{AM}_{1} \mathrm{O} \cong \Delta \mathrm{BM}_{1} \mathrm{O}\) (b) Prove that $\triangle \mathrm{AOB} \cong \triangle \mathrm{BOC} \cong \ldots \triangle \mathrm{EOA}$. (c) Sow that \((\theta / 2)=36^{\circ}\). (d) Prove that \(\mathrm{AM}_{1}=\mathrm{r} \tan 36^{\circ}\). (e) Prove that the perimeter equals \(10 \mathrm{r} \tan 36^{\circ}\).

Chapter 28

Show that the perimeter of an n-sided polygon circumscribed about a circle whose radius has length \(\mathrm{r}\) is \(2 \mathrm{nr} \tan \pi / \mathrm{n}\).

Chapter 28

Find the perimeter of a regular polygon of \(\mathrm{n}\) sides which is circumscribed about a circle of radius length 10 when \(\mathrm{n}\) equals a) \(3 ;\) b) \(4 ;\) c) \(6 ;\) d \(9 ;\) e) \(12 ;\) f) \(18 ;\) g) 36 .

Chapter 28

Show that for a triangle of area \(\mathrm{A}\), and perimeter \(\mathrm{P}\), the radius of the inscribed circle, \(\mathrm{r}\), equals $2 \mathrm{~A} / \mathrm{P}$.

Chapter 28

Show that the perimeter of a square which circumscribes a circle of radius of length \(\mathrm{r}\) is \(8 \mathrm{r}\).

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