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Stage 4 · Geometry

10 problems · AMC 12 late, AIME early · mathsolympiadprep.com

Print or save as PDF The answer key prints on its own page at the end.

1. Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is $m$ times the area of the square. The ratio of the area of the other small right triangle to the area of the square is

   1. A$\frac{1}{2m+1}$
   2. Bm
   3. C1-m
   4. D$\frac{1}{4m}$
   5. E$\frac{1}{8m^2}$

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2. The solutions to the equation $(z+6)^8=81$ are connected in the complex plane to form a convex regular polygon, three of whose vertices are labeled $A,B,$ and $C$. What is the least possible area of $\triangle ABC?$

   1. A$\frac{1}{6}\sqrt{6}$
   2. B$\frac{3}{2}\sqrt{2}-\frac{3}{2}$
   3. C$2\sqrt3-3\sqrt2$
   4. D$\frac{1}{2}\sqrt{2}$
   5. E$\sqrt 3-1$

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3. The perimeter of an isosceles triangle is $24 \mathrm{~cm}$, and a median divides the perimeter into two parts in the ratio 5:3. The length of the base of this triangle is ( $) \mathrm{cm}$.

   1. A7.5
   2. B12
   3. C4
   4. D12 or 4

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4. If the perimeter of the triangle shown is 21 , what is the value of $x$ ?

   1. A3
   2. B7
   3. C8
   4. D13
   5. E16

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5. Four points on a plane, no three of which are collinear, are connected by six line segments. In the figure formed by these line segments, the minimum number of triangles that can be formed is:

   1. A3
   2. B4
   3. C6
   4. D8

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6. A line $l$ is drawn through point $P(2,1)$, intersecting the two coordinate axes at points $A$ and $B$. When the area $S$ of $\triangle A O B$ takes values in $(0,+\infty)$, the number of lines $l$ that can be drawn is ( ) lines.

   1. A2
   2. B3
   3. C2 or 3
   4. D2 or 3 or 4

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7. Given $M(-1,0), N(5, y), P(3,4)$, then the ratio $\lambda$ in which $P$ divides the segment $M N$ is

   1. A$\frac{1}{3}$
   2. B$\frac{1}{2}$
   3. C2
   4. D3

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8. 10) Suppose that, in the circle in the figure, the angle $B A C$ is $35^{\circ}$. Let $C D$ be the diameter passing through $C$, what is the value of $B \widehat{C} D$?

   1. A$35^{\circ}$
   2. B$45^{\circ}$
   3. C$50^{\circ}$
   4. D$55^{\circ}$
   5. Enone of the above

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9. As shown in Figure 4, in rectangle $A B C D$, $A B=5, B C$ $=12$. When rectangle $A B C D$ is folded along the diagonal $A C$ and placed on the table, the area covered by the resulting figure on the table is $\qquad$ .

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10. Calculate the side length of a square inscribed in the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$. (Problem 14, Page 171)

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Answer key — Stage 4 · Geometry

Worked solutions for every problem are on the site, one page per problem.

1. $\frac{1}{4m}$ open
2. $\textbf{(B)}\frac{3}{2}\sqrt{2}-\frac{3}{2}$ open
3. $D$ open
4. $7$ open
5. $B$ open
6. $D$ open
7. $2$ open
8. $55$ open
9. $\frac{2035}{48}$ open
10. $\frac{2 a b}{\sqrt{a^{2}+b^{2}}}$ open

Problems belong to the competitions that set them and are reproduced from open datasets under their licences; every problem page names its source. Free to copy for classroom use.