“You are the average of five people you spend the most time with.” This old saying rings true. In 11th-grade spatial geometry, determining distances can be quite “brain-twisting.” But don’t worry, let “Học Làm” illuminate your path to conquering top scores with ultimate word-based practice tips for 11th-grade geometry distances!
“Decoding” the Distance Matrix: From A to Z
Mastering the theory like knowing a treasure map by heart will help you confidently “win every battle.” So, what foundational knowledge is our “guiding star”?
1. “Universal Key” – Definition and Methods for Determining Distance:
- Distance from a point to a plane: This is the shortest path from the “house” of a point to the “wall” of a plane, always perpendicular to that “wall.”
- Distance between two skew lines: Imagine two lines as two “unrelated” roads; the distance between them is the shortest line segment connecting these two roads, forming right angles with both.
- Distance from a point to a line: The shortest path from the “house” of a point to the “road” of a line, always perpendicular to that “road.”
- Distance between two parallel planes: “Parallel walls,” the distance between them is the line segment connecting the two “walls” and perpendicular to both.
2. “Martial Arts Secrets” – Methods for Solving Distance Problems:
- Auxiliary Line Method: Like “building bridges” as auxiliary roads to shorten the distance, we construct figures to reduce the problem to a known basic distance problem.
- Vector Method: The “ultimate weapon” to help us “attack quickly and win quickly” by setting up the Oxyz coordinate system and calculating with vectors.
- Volume Method: This “fighting style” leverages the relationship between distance and the volume of polyhedra to find the answer.
3. “Practice Drills” – Practice Problems and Detailed Solutions:
“Learning must go hand in hand with practice,” let’s “practice our skills” with some typical practice problems with “Học Làm”!
Problem 1: Given a pyramid S.ABCD with a square base ABCD of side length a, SA perpendicular to the base plane and SA = a√2. Calculate the distance from point A to plane (SBC).
Solution:
Problem 2: In the Oxyz space, given two lines d1: (x-1)/2 = (y+1)/1 = (z-2)/3 and d2: (x+2)/1 = (y-3)/2 = (z+1)/-1. Calculate the distance between d1 and d2.
Solution:
(Detailed solution for Problem 2)
Handy “Pocket” Tips for Distance “Masters”
- Master the Knowledge: “Knowledge is power,” memorize the definitions and theorems related to distance thoroughly.
- Train Geometric Thinking: “Practice makes perfect,” regularly do exercises to develop spatial geometric thinking.
- Use Supporting Tools: “Never stop learning,” utilize spatial geometry drawing software like GeoGebra to visualize more clearly.
“Học Làm” Accompanies You to “Conquer” Every Challenge
On the challenging journey of “passing the dragon gate,” “Học Làm” always accompanies and supports you wholeheartedly. Contact us at 0372888889 or visit us at 335 Nguyen Trai, Thanh Xuan, Hanoi for 24/7 consultation and support.
“Học Làm” believes that with determination and continuous effort, you will reap many successes on your journey to conquer knowledge!