## June 2024 Geometry Regents Exam: Study Notes These notes focus on strategies for maximizing your score on the June 2024 Geometry Regents exam, emphasizing problem-solving techniques and answer elimination. ### Key Exam Strategies * **Show All Work:** Even for Part 1 multiple-choice questions, showing your work can earn partial credit if your answer is incorrect. Treat every multiple-choice question as a short-answer problem to maximize your score. * **Use All Information:** Every numerical detail provided in a problem is usually relevant. Don't dismiss information prematurely. Reread the problem carefully if you're stuck. You might have missed a crucial detail. * **Diagrams Are (Usually) Drawn to Scale:** While the exam states diagrams aren't necessarily drawn to scale, they almost always are. Use this to your advantage to eliminate answers. If you're unsure, try solving the problem *assuming* the diagram is to scale. If you get a valid answer among the choices, it's likely correct. Assuming a diagram is *not* to scale can lead to incorrect answers. * **Utilize the Formula Sheet:** The formula sheet at the end of the exam contains many useful formulas. Familiarize yourself with it before the exam. Practice using the formulas to solve problems before the exam. ### Problem-Solving Techniques & Answer Elimination The following examples illustrate strategies for solving problems and eliminating incorrect answers. #### Problem 1: Triangle Transformations **Problem:** Triangle BR is the image of triangle JOE after a translation. Triangle CA is the image of triangle BR after a line reflection. Which statement is *always* true? * R ≅ T * J ≅ A * JE ≅ RI * OE ≅ AT **Solution:** The order of vertices in the triangle names indicates corresponding vertices. Therefore, only OE ≅ AT is true because the corresponding vertices are in the same order. Let's label the vertices: J - one dash O - two dashes E - three dashes B - one dash R - two dashes I - three dashes C - one dash A - two dashes T - three dashes Now we can see that only OE and AT have corresponding dashes. #### Problem 2: Cross-Sections of a Cylinder **Problem:** A right circular cylinder is cut parallel to its base. What is the shape of the cross-section? **Solution:** A cross-section is a two-dimensional slice. Cutting a cylinder parallel to its base results in a circle. Cone, triangle, and rectangle are impossible cross-sections of a cylinder. Imagine slicing a salami. A horizontal slice gives a circle. #### Problem 3: Rotation of a Hexagon **Problem:** What is the minimum number of degrees a regular hexagon must rotate about its center to carry it onto itself? **Solution:** A hexagon has six sides. A full rotation (360°) divided by the number of sides (6) gives the minimum rotation: 360°/6 = 60°. Imagine rotating a hexagon. It will coincide with itself after rotating 60°, 120°, 180°, 240°, 300°, and 360°. The minimum is 60°. #### Problem 4: Sphere in a Cube **Problem:** A sphere is inscribed in a cube with edges of length 18. What is the volume of the sphere? **Solution:** The diameter of the inscribed sphere is equal to the edge length of the cube (18). The radius is 9. Use the formula for the volume of a sphere: $V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (9^3) = 972\pi$. The diameter of the sphere is 18, so the radius is 9. Substitute into the volume formula: $V = \frac{4}{3}\pi (9)^3 = 972\pi$ #### Problem 5: Similar Triangles **Problem:** Two perpendicular lines intersect at J. Find the length of MJ. **Solution:** Confirm the diagram is drawn to scale. Use similar triangles to set up a proportion and solve for MJ. The triangles are similar by AA similarity. Set up the proportion: $\frac{EA}{AJ} = \frac{HM}{MJ}$. Solve for MJ. #### Problem 6: Perpendicular Lines **Problem:** Find the equation of the line passing through (2,7) and perpendicular to $y = \frac{3}{4}x + 4$. **Solution:** The slope of the given line is $\frac{3}{4}$. The slope of a perpendicular line is the negative reciprocal: $-\frac{4}{3}$. Use point-slope form: $y - 7 = -\frac{4}{3}(x - 2)$. These examples demonstrate how to combine mathematical knowledge with strategic answer elimination to improve your score. Remember to practice consistently and utilize these techniques during the exam.