4: Areas and 3D
BFL Competition Math 2022
Recap of Last Week
Previously, we looked at strategies for solving problems with algebraic manipulations.
Last week’s topic involved relating lengths and angles. Keep in mind that ideas that seem obvious can be really useful!
Review of some areas
Obviously it’s much more important to develop good problem solving skills, but some configurations are helpful to know in order to get them fast and not make avoidable mistakes.
The Area Lemma
Consider triangle ABC.
Let D be some point on side BC. What happens to the area of [ABD] as we slide it around side BC?
Draw a segment connecting A and D, and put a point P anywhere on AD. What happens to the area of [PBC] if we slide P around?
Area Lemma: [ABD]/[ACD] = BD/CD, [ABC]/[PBC] = AD/PD
Using the area lemma - examples
Consider triangle ABC with area 60. Let D lie on side BC such that BD/DC = ½. Find the area of triangle ABD.
Let F be the midpoint of AD in this diagram. Now find the area of triangle BFD.
Prove that the medians of a triangle divide it into six equal areas.
Problem (2021 AoPS Practice AMC)
On triangle ABC, points R, S, T are marked on AB, BC, CA respectively such that AR = ⅓ AB, BS = ¼ BC, and CT = ⅕ CA. If the area of triangle ABC is 120, then what is the area of triangle RST?
Area Manipulation Strategies
Cutting up an area into shapes that are easier to handle is helpful if we see a shape to go for. (look for right angles, sectors, triangles, etc.)
When we see areas that overlap, we have to subtract out the overlapping region since it gets counted twice. (like a Venn diagram)
Don’t forget you can use algebra on some formulas if you see that it’ll help!
Problem (2021 AMC 10B)
The figure is constructed from 11 line segments, each of which has length 2. The area of pentagon ABCDE can be written as , where m and n are positive integers. What is m+n?
Review of 3D knowledge
The volume of a box is xyz, where x, y, and z are its dimensions.
The volume of any prism is the area of the base times its height.
The volume of a cone or pyramid is ⅓ times the area of its base times its height.
The volume of a sphere is 4/3 πr3
For some reason, AMC loves cone problems (there’s pretty much one every year), so let’s do one
Problem (2020 AMC 10B)
A three-quarter sector of a circle of radius 4 inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?
3D Problem Solving Strategies
Like with 2D, look for simpler figures that you can cut up, overlap, etc.
A very important tactic is to work with cross-sections.
Cross sections can help you see lengths and angles and actually be able to draw a diagram since 2D is easier to work with.
For example, turning cones into triangles, spheres into circles, etc.
As a last resort, make it a coordinate system if you see things like right angles
Problem (2013 AMC 10A)
Six spheres of radius 1 are positioned so that their centers are at the vertices of a regular hexagon of side length 2. The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of the eighth sphere?
Problem (2021 AMC 10A)
What is the volume of tetrahedron ABCD with edge lengths AB=2, AC=3, AD=4, BC=√13, BD=2√5, and CD=5?
Recap
Just like with algebra, look for ways to move things around!
For instance, cutting things up, subtracting out overlaps, etc.
The most important 3D strategy is to take cross sections.
Thank you for coming!
Slides and problems will be posted in the next few days, reach out to dengr557@gmail.com with any questions