☰ Contents · Geometry

Multiplying a vector by a number and applying vectors

Lessons 43–44 · 2 lessons · A. A. Rahimqoriyev, M. A. To‘xtaxo‘jayeva. Geometry, Grade 8 (textbook for general secondary schools), revised 3rd edition. Yangiyo‘l Poligraf Servis, Tashkent, 2014
44

Using vectors to solve problems

Textbook: pp. 139–141
GoalKnow how to use vectors to solve geometric problems and prove theorems.
New words
midpoint of a segment · kesma o‘rtasimidline · o‘rta chiziqvector method · vektor usuliexpressing a vector · vektorni ifodalash
Explanation

We solve problems with vectors in three stages: write the condition in vector form, transform it with vector algebra, and interpret the result in geometric language. First result: if C is the midpoint of AB and O is any point, then →OC = ½(→OA + →OB). To prove it, add →OC = →OA + →AC and →OC = →OB + →BC; since →AC + →BC is the zero vector, 2→OC = →OA + →OB. Second result, the midline theorem: if E and F are the midpoints of AB and AC, then →EF = →AF − →AE = ½(→AC − →AB) = ½→BC. Hence EF ∥ BC and EF = ½BC.

Worked examples
If C is the midpoint of AB, then →AC = ½→AB; if AB = 22, then |→AC| = 11.
In triangle ABC with BC = 18 the midline EF satisfies EF ∥ BC and EF = 18 : 2 = 9.
Class activity

“Find the middle”: draw a triangle, mark the midpoints of two sides, and measure the midline and the third side to compare.

Practice
1
M is the midpoint of AB and AB = 22. What is |→AM|?
2
One side of a triangle is 30. How long is the midline parallel to it?
3
C is the midpoint of AB and |→OA + →OB| = 18. What is |→OC|?
4
Why does →EF = ½→BC give both EF ∥ BC and EF = ½BC?