Geometry — Testing-Out Examination
A student who passes this examination has demonstrated mastery of the Common Core Geometry standards (CCSS-M conceptual categories G-CO, G-SRT, G-C, G-GPE, G-GMD, G-MG, S-CP) and is eligible to advance to Algebra 2.
Instructions
- Time limit: 120 minutes total. Part I (no-calculator) is intended to take about 45 minutes; Part II (calculator-permitted) the remaining 75 minutes.
- Show all work. Proofs must be written in two-column or paragraph form with reasons. Constructions (when asked) must list each step.
- Diagrams may be sketched freehand but should be reasonably accurate. Do not assume relationships beyond what is given.
- Round only when the problem says so; otherwise express exact values.
- Use of straightedge / protractor / compass on Part I is permitted; calculators are not.
Triangle \( ABC \) has vertices \( A(1, 2),\ B(5, 2),\ C(3, 5) \). Triangle \( A'B'C' \) has vertices \( A'(-2, 1),\ B'(-2, 5),\ C'(-5, 3) \).
- Show that \( \triangle ABC \cong \triangle A'B'C' \) by computing the lengths of all three pairs of corresponding sides. Use the distance formula and show your work.
- Describe a sequence of rigid motions that maps \( \triangle ABC \) onto \( \triangle A'B'C' \). Be specific (e.g., "rotation 90° counterclockwise about the origin, then translation by \(\langle a, b \rangle\)"). Verify the image of at least two vertices after each motion.
- Is the sequence in (b) unique? Give a one-sentence justification.
In \(\triangle ABC\), point \(D\) lies on \(\overline{AB}\) and point \(E\) lies on \(\overline{AC}\) with \(\overline{DE} \parallel \overline{BC}\). Given \( AD = 6 \), \( DB = 9 \), and \( AE = 4 \).
- Justify why \(\triangle ADE \sim \triangle ABC\). Cite the appropriate similarity postulate or theorem.
- Find \(EC\). Show the proportion you set up.
- Given \(BC = 25\), find \(DE\).
- The area of \(\triangle ADE\) is \(16\) cm\(^2\). Find the area of \(\triangle ABC\). Explain in one sentence why area scales as the square of the linear ratio.
Given: \(ABCD\) is a parallelogram with diagonal \(\overline{AC}\).
Prove: \(\triangle ABC \cong \triangle CDA\).
Provide a two-column proof with at least six statements and matching reasons. You may use any properties of parallelograms (opposite sides parallel; opposite sides congruent; alternate-interior-angles theorem; reflexive property) but cite each property by name.
| Statement | Reason |
|---|---|
| 1. | 1. |
| 2. | 2. |
| 3. | 3. |
| 4. | 4. |
| 5. | 5. |
| 6. | 6. |
- State the chord–chord power-of-a-point theorem: if chords \(\overline{AB}\) and \(\overline{CD}\) intersect at \(P\) inside a circle, then \(AP \cdot PB = CP \cdot PD\). Given \(AP = 4\), \(PB = 9\), \(CP = 6\), find \(PD\).
- An inscribed angle of a circle subtends an arc of measure 80°. Find the measure of the inscribed angle. State which theorem you used.
- From an external point \(T\), a tangent of length 12 and a secant are drawn to a circle. The secant intersects the circle at points \(A\) and \(B\) with \(A\) the nearer intersection and \(TA = 8\). Find \(AB\). Show your work using the tangent–secant relationship \( (TA + AB) \cdot TA = (\text{tangent})^2 \).
- Find the equation of the circle that passes through \((0,0)\), \((6,0)\), and \((0,8)\). Use either the general form \(x^2 + y^2 + Dx + Ey + F = 0\) and three equations, or recognize a special triangle. Show your work and write your final answer in standard form \((x - h)^2 + (y - k)^2 = r^2\).
A surveyor stands at point \(P\) on level ground and measures the angle of elevation to the top of a building as \(38°\). She walks 50 feet directly toward the building and measures a new angle of elevation of \(54°\). Let \(h\) be the height of the building and \(d\) be her distance from the building's base after walking.
- Set up two trigonometric equations relating \(h\), \(d\), and the given angles.
- Solve the system for \(h\) and \(d\). Round each to the nearest tenth of a foot. Show every algebraic step.
- Verify your answer by checking that \(\tan(54°) \cdot d \approx h\) within rounding tolerance.
- State the height of the building rounded to the nearest foot.
Let \(A = (-2, 1)\), \(B = (3, 4)\), \(C = (5, 0)\), \(D = (0, -3)\).
- Compute the lengths of all four sides \(\overline{AB}\), \(\overline{BC}\), \(\overline{CD}\), \(\overline{DA}\). Show your work.
- Compute the lengths of both diagonals \(\overline{AC}\) and \(\overline{BD}\).
- Compute the slopes of all four sides. Identify any pair of sides that is parallel and any pair that is perpendicular.
- Based on (a)–(c), classify quadrilateral \(ABCD\) as most specifically as possible (parallelogram, rectangle, rhombus, square, or none of these). Justify with the appropriate property or properties.
A grain silo consists of a right circular cylinder of radius 8 ft and height 30 ft, capped by a hemispherical dome of radius 8 ft.
- Find the total volume of the silo (cylinder + dome) in cubic feet. Express the exact value (in terms of \(\pi\)) and the decimal value to the nearest cubic foot.
- Find the total external surface area (lateral surface of the cylinder + curved surface of the hemisphere; do not include the floor). Express both exactly and to the nearest square foot.
- The silo is filled with grain of mass density 50 lb/ft\(^3\). Find the total mass of grain in pounds. Round to the nearest pound.
- Suppose the silo is partially filled to a depth of 25 ft, measured from the floor (so only a portion of the cylinder is filled, none of the dome). What percent of the silo's total capacity is filled? Round to the nearest 0.1%.
A right square pyramid has height 12 cm and a square base of side 6 cm. A right rectangular pyramid has the same height 12 cm and a rectangular base of dimensions 4 cm by 9 cm.
- State Cavalieri's principle in your own words in one sentence.
- Compute the volume of each pyramid using \(V = \tfrac{1}{3} \cdot \text{base} \cdot \text{height}\). What do you observe about the two volumes, and how does Cavalieri's principle explain it?
- For the square pyramid, take a horizontal cross-section at height 4 cm above the base. Use a similarity argument to find the side length of the cross-sectional square; then state its area.
A beverage can is approximately a right circular cylinder of radius \(r = 3\) cm and height \(h = 12\) cm.
- Compute the volume of the can in cubic centimeters. Express both exactly (in terms of \(\pi\)) and as a decimal to the nearest cubic cm.
- Compute the total surface area (top + bottom + lateral). Express both exactly and as a decimal to the nearest square cm.
- Suppose the manufacturer doubles the radius (to \(r = 6\)) but keeps the height the same. By what factor does the volume change? By what factor does the lateral surface area change? Justify each in one sentence.
- A label is wrapped exactly around the lateral surface of the original can (radius 3, height 12). Find the dimensions and area of the rectangular label.
A school board member is reviewing data on athletic participation by grade level to determine whether varsity-sport participation differs between Juniors and Seniors. Two hundred high-school students were classified by grade level (Junior or Senior) and by whether they participate in a varsity sport. The results are shown.
| Varsity sport (S) | No varsity sport (\(S^c\)) | Total | |
|---|---|---|---|
| Junior (J) | 42 | 58 | 100 |
| Senior (\(J^c\)) | 56 | 44 | 100 |
| Total | 98 | 102 | 200 |
- If a student is selected at random, find \(P(S)\) and \(P(J)\).
- Find \(P(S \cap J)\), the probability that a randomly chosen student is a Junior who plays a varsity sport.
- Find \(P(S \mid J)\), the probability that a student plays a varsity sport given that the student is a Junior. Show the conditional-probability formula explicitly. Then compute \(P(S \mid J^c)\). Compare the two and comment in one sentence on whether playing a varsity sport appears associated with grade level.
- Test independence: are events \(S\) and \(J\) independent? Justify by checking either \(P(S \cap J) \stackrel{?}{=} P(S) \cdot P(J)\) or \(P(S \mid J) \stackrel{?}{=} P(S)\). Show your computation explicitly.
- Two students are selected at random without replacement. Find the probability that both play a varsity sport. (Use the multiplication rule for dependent events.)