Medians of a Triangle | Problems 1-10 | Problem Solving | Volchkevich | Geometry Lessons Grade 7-8
1. Two medians in a triangle are congruent. Prove that it is isosceles.
*2. Can the statement in the previous problem be proven without Euclid's fifth postulate?
3. (Property of the medians of a triangle.) Prove that all medians of a triangle intersect at a single point and are divided by it in the ratio 2:1, counting from the vertex. Use the provided diagram for proof.
4. Two medians of a triangle are perpendicular. Find the ratio of its third median to the corresponding side.
5. On the extension of side AC of triangle ABC, point K is taken such that CK = AC. Point M is the midpoint of AB. In what ratio does
line MK divide side BC?
6. A point on a leg of a right triangle is connected to
one of its vertices and the midpoint of the hypotenuse. It turns out that
the angles marked in the diagram are equal. In what ratio does this point divide the leg?
7. In trapezoid ABCD, point M is the midpoint of side CD.
On segment AM, point O is chosen such that AO : OM = 2 : 1. Line BO
intersects base AD at point E. Prove that segment AE
is equal to the midline of the trapezoid.
*8. Point E is the midpoint of side CD of parallelogram ABCD,
point K lies on side AD. On segment BE, point O is chosen such that
BO = 2 · OE. Line OK intersects side BC at point M. Find
the ratio CM : AK.
9. Prove that a new triangle can always be formed from the medians of an arbitrary triangle such that its sides are parallel to the given medians.
10. A new triangle is formed from the medians of a given triangle. An arbitrary median is drawn in it. Prove that
it is 3/4 of one side of the original triangle.