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Stage 2 · Geometry

10 problems · · mathsolympiadprep.com

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

1. Two identical smaller cubes are stacked next to a larger cube. Each of the two smaller cubes has a volume of 8. What is the volume of the larger cube?

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2. The entire exterior of a solid $6 \times 6 \times 3$ rectangular prism is painted. Then, the prism is cut into $1 \times 1 \times 1$ cubes. How many of these cubes have no painted faces?

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3. In the diagram, $PQRS$ is a square and $PTQ$ is an equilateral triangle.

   What is the measure of $\angle TPR$?

   1. A$90^{\circ}$
   2. B$105^{\circ}$
   3. C$120^{\circ}$
   4. D$150^{\circ}$
   5. E$75^{\circ}$

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4. In $\triangle ABC$, points
   $E$ and $F$ are on $AB$ and $BC$, respectively, such that $AE=BF$ and $BE=CF$.

   If $\angle BAC = 70\degree$, the
   measure of $\angle ABC$ is

   1. A$40\degree$
   2. B$50\degree$
   3. C$60\degree$
   4. D$70\degree$
   5. E$30\degree$

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5. Points $A, B, C$, and $D$ are on a line in that order. The distance from $A$ to $D$ is 24. The distance from $B$ to $D$ is 3 times the distance from $A$ to $B$. Point $C$ is halfway between $B$ and $D$. What is the distance from $A$ to $C$?

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6. Square $WXYZ$ has area 9 and $W$ is at the centre of a circle.

   If $X$ and $Z$ are on the circle, the area of the circle is

   1. A$3\pi$
   2. B$6\pi$
   3. C$9\pi$
   4. D$18\pi$
   5. E$81\pi$

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7. In the diagram, $PQRS$ is a square and $M$ is the midpoint of $PQ$. The area of triangle $MQR$ is 100. The area of the square $PQRS$ is

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8. What is the measure of the largest angle in $\triangle P Q R$?

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9. In the diagram, $PQRS$ is a rectangle. Also, $\triangle STU$, $\triangle UVW$ and $\triangle WXR$ are congruent.

   What fraction of the area of rectangle $PQRS$ is shaded?

   1. A$\frac{3}{7}$
   2. B$\frac{3}{4}$
   3. C$\frac{1}{2}$
   4. D$\frac{3}{5}$
   5. E$\frac{2}{3}$

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10. In the diagram, $\angle PQR$ is a straight angle.

    The value of $x$ is

    1. A$100$
    2. B$140$
    3. C$50$
    4. D$120$
    5. E$320$

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Answer key — Stage 2 · Geometry

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

1. $64$ open
2. $16$ open
3. $B$ open
4. $A$ open
5. $15$ open
6. $C$ open
7. $D$ open
8. $108^{\\circ}$ open
9. $C$ open
10. $B$ 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.