Revision Sheet 3

Revision sheet 3 I.

1. 2. 3. 4. 5.

Consider two concentric circles C1 (O, r ) and C 2 (O, R) such that r < R . Let M be a point of circle C1 . The tangent at M to circle C1 cuts circle C 2 in A and B. Show that M is the midpoint of [AB]. Draw the tangents [AE) and [BF) of the circle C1 , where E and F are the points of tangency. Show that AE=BF. (AE) and (BF) intersect at N. show that the triangle NAB is isosceles. Deduce that the points M, O, and N are collinear. Show that (AB) and (EF) are parallel. In this question, suppose that angle ENF measures 60  and NF = 3 2 cm. Calculate NO.

II.

ABC is a triangle right at A. [AH] is the altitude relative to [BC]. M is the midpoint of [HC] and N is the midpoint of [AH]. Show that N is the orthocenter of triangle ABM.

III.

ABCD is a rectangle such that AB=6cm and BC=3cm. E is the midpoint of [DC]. a) Show that triangle AEB is right at E. b) Show that (AE) is tangent to a circle ( C ) whose center and radius are to be determined. ( Calculate the radius)

IV.

( C ) is a semi circle of center O, diameter [AB] and radius 4cm. E is a point of this semi-circle such that EAˆ B = 30  . The tangent at E to ( C ) cuts the tangent at A to ( C ) at point M. a) What is the nature of triangle MEA? b) (MO) cuts [AE] at point I. Show that I belongs to a circle of diameter [AO]. c) Calculate the lengths: MO; AI;MI. d) Draw the circle ( C’) of diameter [EB]. What is the position of line ( AE) with respect to this circle?

V.

Consider a circle C(O;4cm). A straight line (d) passing through O cuts the circle in A and B. Let M be the symmetric of O with respect to A. Draw (MT) tangent to the circle (C) at T. a) Draw the figure. b) Show that TMˆ O = 30  c) Calculate the lengths: MT; AT; TB.

d) The straight line (d’) perpendicular to (d) at B cuts the tangent (MT) at E. i) What is the nature of triangle TBE? ii) (TB) cuts (OE) at I. What is the position of (TB) with respect to the circle of center O and radius 2cm? VI.

( C ) is a circle of center O and radius 3cm and diameter [AB]. C is a point on [AB) such that BC=3cm. Let D be the point of tangency drawn from C to the circle. Draw from point A a line parallel to (OD) to cut [CD) at point E. 1) Draw the figure. 2) What is the nature of triangle DOC? 3) Calculate EC and AE. 4) Show that the circle (C ;

9 3 ) is tangent to (AE) at point E. 2

VII. [AB] is a diameter of the circle whose diameter is 4cm. [AC] is a chord such that BAˆ C = 30  . D is a point outside the circle. The tangent from D cuts the circle at point C. Calculate BD.