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Geometry and Measurement

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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