Geometry and Trigonometry

Chief Editor

Mar 27, 2024

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The measure of angle $R$ is $\frac{2\pi}{3}$ radians. The measure of angle $T$ is $\frac{5\pi}{12}$ radians greater than the measure of angle $R$. What is the measure of angle $T$, in degrees?

Difficulty: Medium

75

120

195

390

Triangle $ABC$ is similar to triangle $DEF$, where $A$ corresponds to $D$ and $C$ corresponds to $F$. Angles $C$ and $F$ are right angles. If $\tan(A)=\sqrt{3}$ and $DF=125$, what is the length of $\overline{DE}$?

Difficulty: Hard

125\frac{\sqrt{3}}{3}

125\frac{\sqrt{3}}{2}

125\sqrt{3}

$250$

The equation $x^{2}+\left(y-1\right)^{2}=49$ represents circle A. Circle B is obtained by shifting circle A down $2$ units in the $xy$-plane. Which of the following equations represents circle B?

Difficulty: Hard

(x-2)^{2}+(y-1)^{2}=49

x^{2}+(y-3)^{2}=49

(x+2)^{2}+(y-1)^{2}=49

x^{2}+(y+1)^{2}=49