Troy School District · Mathematics · Geometry · Form B
Standards Drill · 120 minutes · 100 points

Geometry — Standards-Aligned Skills Examination

A calculation-focused, standards-walkthrough exam covering every CCSS-M Geometry conceptual category (G-CO, G-SRT, G-C, G-GPE, G-GMD, G-MG, S-CP). Each problem header cites the standard(s). Form B complements Form A (which is more applied / problem-solving and proof-heavy).

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Instructions

1. G-CO.A · Rigid motions on the coordinate plane 8 points

Triangle \(\triangle ABC\) has vertices \(A(2, 1)\), \(B(5, 1)\), \(C(5, 4)\). Apply each transformation to \(\triangle ABC\) and state the new coordinates of the image.

  1. Translation by the vector \(\langle -3, 4\rangle\).
  2. Reflection across the \(x\)-axis.
  3. Reflection across the \(y\)-axis.
  4. Reflection across the line \(y = x\).
  5. Rotation 90° counterclockwise about the origin.
  6. Rotation 180° about the origin.
  7. Dilation centered at the origin with scale factor 3.
  8. Dilation centered at \((2, 1)\) with scale factor 2.
2. G-CO.B–C · Angle relationships 10 points
  1. Two angles are complementary. One measures \( (3x + 7)° \), the other \( (2x - 12)° \). Find \(x\) and each angle.
  2. Two angles are supplementary. One measures \( (5x - 4)° \), the other \( (2x + 19)° \). Find \(x\) and each angle.
  3. In a triangle, the three angles measure \( (x + 20)°,\ (2x - 5)°,\ (x + 25)° \). Find \(x\) and each angle. Verify the sum is 180°.
  4. Parallel lines \(\ell\) and \(m\) are cut by a transversal. Two corresponding angles measure \( (4x + 15)° \) and \( (6x - 25)° \). Find \(x\).
  5. Same configuration: alternate-interior angles measure \( (3x + 22)° \) and \( (5x - 8)° \). Find \(x\).
  6. Same configuration: co-interior (consecutive interior) angles measure \( (2x + 30)° \) and \( (3x - 10)° \). Find \(x\) (these sum to 180°).
  7. An interior angle of a regular polygon measures 144°. How many sides does the polygon have? Use the formula \( \dfrac{(n-2) \cdot 180}{n} \).
  8. Find the sum of the interior angles of a 12-gon.
3. G-SRT.A · Similarity and scale 10 points
  1. \(\triangle ABC \sim \triangle DEF\) with \(AB = 8\), \(DE = 12\), \(BC = 10\). Find \(EF\). Show the proportion.
  2. Two similar polygons have a linear scale factor of \(2 : 5\). Find:

    (i) the ratio of perimeters    (ii) the ratio of areas    (iii) the ratio of volumes (for similar 3-D solids)

  3. The areas of two similar triangles are in the ratio \(9 : 25\). Find the ratio of corresponding side lengths. Then find the ratio of the perimeters.
  4. In \(\triangle ABC\), \(\overline{DE} \parallel \overline{BC}\) with \(D\) on \(\overline{AB}\) and \(E\) on \(\overline{AC}\). \(AD = 5\), \(DB = 7\), \(AE = 4\). Find \(EC\).
  5. A 6-ft tall person casts a 9-ft shadow at the same time a building casts a 75-ft shadow. Find the height of the building.
  6. On a map drawn at scale \(1 : 24{,}000\), two cities are 8 inches apart. Find the actual distance in miles. (1 mile = 63{,}360 in.)
4. G-SRT.C · Right-triangle trigonometry 12 points
  1. In right triangle \(\triangle ABC\) with \(\angle C = 90°\), legs \(a = 5\) and \(b = 12\), find:

    (i) the hypotenuse \(c\)    (ii) \(\sin A,\ \cos A,\ \tan A\)    (iii) \(\angle A\) (round to nearest 0.1°)    (iv) \(\angle B\)

  2. In a right triangle, an acute angle is 35° and the opposite side is 9. Find:

    (i) the adjacent side    (ii) the hypotenuse

  3. In a right triangle, the hypotenuse is 20 and one leg is 12. Find the other leg and both acute angles.
  4. Use exact values for special right triangles:

    (i) In a 30-60-90 triangle, if the shorter leg is 7, find the longer leg and hypotenuse exactly.
    (ii) In a 45-45-90 triangle, if a leg is 6, find the hypotenuse exactly.

  5. From the top of a 80-ft cliff, the angle of depression to a boat at sea is 22°. How far is the boat from the base of the cliff? Round to the nearest foot.
  6. A ramp makes an 11° angle with horizontal and rises 3 ft. How long is the ramp (slope distance)? Round to the nearest 0.1 ft.
5. G-C · Circle measurements 10 points
  1. A circle has radius 7 cm. Find:

    (i) circumference    (ii) area    (iii) length of a 60° arc    (iv) area of a 60° sector

  2. A central angle of 90° subtends an arc of length 5π. Find the radius of the circle.
  3. The area of a sector is \(12\pi\) cm\(^2\) and the radius is 6 cm. Find the central angle in degrees.
  4. An inscribed angle subtends an arc of 130°. Find the inscribed angle.
  5. From an external point, two tangent segments to a circle have lengths \(2x + 1\) and \(x + 7\). Find \(x\) (use the equal-tangents theorem).
  6. Two chords intersect inside a circle. One chord is divided into segments of length 4 and 9. The other chord is divided into segments of length 6 and \(x\). Find \(x\) (use the chord-chord product).
6. G-GPE · Coordinate geometry 10 points
  1. Find the distance, midpoint, and slope of the segment with endpoints \( (-3, 5) \) and \( (4, -1) \). Show each formula.
  2. Find the equation of the line through \( (-2, 3) \) parallel to \( y = -\tfrac{2}{3} x + 5 \).
  3. Find the equation of the line through \( (4, -1) \) perpendicular to \( y = \tfrac{1}{4} x + 2 \).
  4. Find the perpendicular bisector of the segment with endpoints \((1, 2)\) and \((7, 6)\). Show your work.
  5. Find the area of the triangle with vertices \(A(0, 0)\), \(B(6, 0)\), \(C(4, 5)\). Use the base-and-height formula.
  6. Find the perimeter of the quadrilateral with vertices \( (1, 1),\ (5, 1),\ (5, 4),\ (1, 4) \) using the distance formula.
7. G-GPE.A.1 · Equations of circles and parabolas 8 points
  1. Write the equation of a circle in standard form with center \( (3, -4) \) and radius 5.
  2. Convert each equation to standard form by completing the square; state center and radius:

    (i) \( x^2 + y^2 - 6x + 4y - 12 = 0 \)    (ii) \( x^2 + y^2 + 8x - 2y + 1 = 0 \)

  3. The endpoints of a diameter of a circle are \((1, 2)\) and \((7, 10)\). Find the equation of the circle. Show how you find center (midpoint) and radius (half the diameter).
  4. Write the equation of a parabola with vertex \((1, -2)\) opening upward and passing through \((3, 6)\).
8. G-GMD · Volume and surface area of solids 12 points
  1. Compute the volume of each:

    (i) rectangular prism, dimensions 4 × 6 × 10    (ii) cube, edge length 5    (iii) cylinder, radius 3, height 8    (iv) cone, radius 4, height 9    (v) sphere, radius 6    (vi) square pyramid, base side 8, height 12

  2. Compute the surface area of each:

    (i) cube, edge 5    (ii) cylinder, radius 3, height 8 (lateral + two circles)    (iii) sphere, radius 6

  3. Find the missing dimension:

    (i) A cylinder of radius 5 has volume \(150\pi\) cm\(^3\). Find its height.
    (ii) A cone of radius \(r\) and height 12 has volume \(64\pi\). Find \(r\).
    (iii) A sphere has volume \(\dfrac{500\pi}{3}\). Find its radius.

9. G-MG · Modeling with geometry 10 points
  1. A rectangular concrete pad is 12 ft × 8 ft × 0.5 ft. Concrete weighs 150 lb/ft\(^3\). Compute the weight of the pad.
  2. Aluminum has density 2.7 g/cm\(^3\). A solid aluminum sphere has radius 4 cm. Compute the mass.
  3. A cylindrical water tank has radius 5 ft and height 12 ft. Find the capacity in gallons (1 ft\(^3\) ≈ 7.48 gal). Round to the nearest gallon.
  4. A road's grade (incline) is 6%, meaning vertical rise is 6% of horizontal distance. If a road climbs 24 ft vertically, find the horizontal distance traveled. Find the road length (slope distance).
  5. A triangular plot of land has vertices that, in coordinates, are at \((0, 0)\), \((300, 0)\), \((180, 240)\) (units in feet). Find the area in acres (1 acre = 43{,}560 ft\(^2\)). Round to two decimals.
10. S-CP · Conditional probability and independence 10 points

A standard 52-card deck. A card is drawn at random. Let \(K\) be the event "card is a King" and \(H\) be the event "card is a Heart."

  1. Compute \(P(K),\ P(H),\ P(K \cap H),\ P(K \cup H)\).
  2. Compute \(P(K \mid H)\) and \(P(H \mid K)\). Show the conditional formula \(P(A \mid B) = P(A \cap B) / P(B)\).
  3. Are \(K\) and \(H\) independent? Justify by checking \(P(K \cap H) \stackrel{?}{=} P(K) \cdot P(H)\).
  4. Two cards are drawn without replacement. Compute the probability that both are Aces. Show the dependent-trials product.
  5. A two-way table classifies 100 students by gender (M, F) and music preference (Pop, Rock):
    PopRockTotal
    M222850
    F341650
    Total5644100
    Compute: (i) \(P(\text{Pop})\); (ii) \(P(F \mid \text{Pop})\); (iii) \(P(\text{Pop} \mid F)\); (iv) Are gender and preference independent? Justify.