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# SAT Geometry Perfect Score

In the above diagram, quadrilateral $$ABDE$$ has two right angles $$\angle B$$ and $$\angle D,$$ and $$C$$ is a point on $$\overline{BD}.$$ The lengths of some of the line segments are $\begin{array}&\lvert\overline{AB}\rvert=m, &\lvert\overline{BC}\rvert=n, &\lvert\overline{CA}\rvert=5, &\lvert\overline{AE}\rvert=13, &\lvert\overline{EC}\rvert=mn, \end{array}$ where $$m$$ and $$n$$ are integers such that $$m<n.$$ Then what is $\lvert\overline{CD}\rvert+\lvert\overline{DE}\rvert?$

Note: The above diagram is not drawn to scale.

(A)$$\ \ 14$$
(B)$$\ \ \frac{84}{5}$$
(C)$$\ \ 17$$
(D)$$\ \ 18$$
(E)$$\ \ \frac{94}{5}$$

If the following three geometric shapes have the same surface area, what is $$h^2a^2?$$

$$\begin{array}{r r l} & \text{I.} & \text{a sphere with radius } 1\\ & \text{II.} & \text{a right circular cone with base radius } 1 \text{ and height } h\\ & \text{III.} & \text{a regular tetrahedron with edge length } a\\ \end{array}$$

(A)$$\ \ 32\sqrt{2}$$
(B)$$\ \ 16\pi$$
(C)$$\ \ 32\sqrt{3}$$
(D)$$\ \ \frac{32\sqrt{3}\pi}{3}$$
(E)$$\ \ 16\sqrt{2}\pi$$

Four line segments $$\overline{AB}, \overline{CD}, \overline{ED},$$ and $$\overline{FG}$$ intersect at three points $$H, J,$$ and $$K,$$ as shown in the above diagram. If $\begin{array}&\angle AKC= \angle BHF, &\angle GJK= \angle EHF, &\angle JDH=40 ^\circ,\end{array}$ what is the measure (in degrees) of $$\angle BHD?$$

Note: The above diagram is not drawn to scale.

(A)$$\ \ 60$$
(B)$$\ \ 65$$
(C)$$\ \ 70$$
(D)$$\ \ 75$$
(E)$$\ \ 80$$

Line segments $$\overline{AB}$$ and $$\overline{CD}$$ are parallel. The measures (in degrees) of some angles are as follows: $\begin{array} &\angle BEF = x, &\angle FHD = x^2, &\angle FGD = 5x. \end{array}$ If $$\angle EFG=2\angle GFH,$$ what is $$x$$ in degrees?

(A)$$\ \ 6$$
(B)$$\ \ 6.5$$
(C)$$\ \ 7$$
(D)$$\ \ 7.5$$
(E)$$\ \ 8$$

Line segments $$\overline{AB}$$ and $$\overline{CD}$$ are parallel and $$EFGHI$$ is a regular pentagon. If $$\angle BEF=7\angle AEI,$$ what is the measure of $$\angle CGH?$$

Note: The above diagram is not drawn to scale.

(A)$$\ \ 24^\circ$$
(B)$$\ \ 27^\circ$$
(C)$$\ \ 30^\circ$$
(D)$$\ \ 33^\circ$$
(E)$$\ \ 36^\circ$$

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