Download MathTime. GEOMETRY. 2

MathTime. GEOMETRY. 2
Special parallelograms
A rectangle is a quadrilateral with four right angles.
B
C
A
D
A rhombus is a quadrilateral with four equal sides.
1. Prove that the diagonals of a rectangle are equal: AC = BD.
2. Prove that if the diagonals of a parallelogram are equal then it is a rectangle.
3. Prove that the diagonals of a rhombus are perpendicular
and bisect the angles.
4. Prove that if the diagonals of a parallelogram bisect the
angles then it is a rhombus.
5. Prove that if the diagonals of a parallelogram are perpendicular then it is a rhombus.
Special triangles
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leg
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A triangle is called right if it has a
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se
right angle.
leg
A triangle is called isosceles if it
Right triangle Isosceles triangle Equilateral triangle
has two equal sides.
A triangle is called equilateral if it has three equal sides.
Remarkable lines in a triangle
A median is a segment joining a vertex with the midpoint of the opposite side.
A height (altitude) is a segment through a vertex perpendicular to the opposite side.
An angle bisector is a segment that divides the angle into two equal parts.
6. B
D
A
Prove that the median AD of a triangle
ABC equals half of the side BC if and only
if A is a right angle.
C
7. Prove that the leg opposite to an angle of 30◦ in a right triangle
is equal to half of the hypotenuse.
8. Prove that a triangle is isosceles if and only if the angles adjacent
to its base are equal.
9. Prove that in isosceles triangle the median, the height, and bisector drawn toward its base coincide.
10. Compute the angles of an equilateral triangle.
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Remarkable points in a triangle
Theorem 1. The perpendicular bisectors of a triangle intersect
at one point. This point is the center of a circle (circumcircle)
that circumscribes the triangle.
It is called the center (circumcenter) of the triangle.
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mc i r c l
cu
e
r
i
Theorem 2. The heights of a triangle intersect at
one point.
It is called the orthocenter of the triangle.
c
in
Theorem 3. The angle bisectors of a triangle intersect at one
point. This point is the center of a circle (incircle) inscribed
into the triangle.
It is called the incenter of the triangle.
i r c le
Theorem 4. (a) The medians of a triangle intersect at one point.
It is called the barycenter (centroid) of the triangle.
(b) The barycenter divides each median in a ratio 2/1.
11. In a ∆ABC the angle m∠ABC = 120◦ , AB = BC = 4. Find the radius (circumradius) of the
circumcircle.
12. A diameter and chord are drawn through a point on a circle. The length of the chord is equal to
the radius. Find the angle between the diameter and the chord.
13. Two chords are drawn through a point on a circle. The length of each of them is equal to the
radius. Find the angle between the chords.
14. Two perpendicular diameters are drawn in a circle of radius R. A point on the circle is projected
on the diameters. Find the distance between the projections.
C
15. A chord AB is drawn in a circle centered at O. It is extended out
of the circle on the distance BC which is equal to the radius of
the circle. A straight line is drawn through the points C and O.
Let D be the point of its intersection with the circle that does
not lie on the segment CO. Prove that m∠AOD = 3m∠ACD.
B
A
O
D
C
16.
B
A circle cuts off two chords of equal lengths, BC = DE, on two lines
intersecting at point A. Prove that AC = AE, and AB = AD.
A
D
E
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