2013 first semester exam

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CCS Mathematics 23 Dec. 2013
Class of G8 Exam of 𝟏 𝒔𝒕
semester Duration : 2h
Name :…………………………………..
I. (2 points)
Given the two circles C(A ;r) and C’(B ;R).
1. Study the relative position of these circles if r= 3 cm R= 4 cm and AB= 7 cm.
2. Study the relative position of these circles if r= 2 cm , R= 4 cm and AB= 2 cm.
II. (2 points)
In the following figure we have the angle 𝐽𝐼̂ 𝐾 = 45° in the circle
C( O ; 2 cm).
1. Prove that OJK is a right isosceles.
2. Calculate the area of the circular sector 𝐽𝑂̂ 𝐾.
3. Deduce the area of the hachured part.
III. (4 points)
1. The angles of the quadrilateral ABCD are given by : 𝐴̂ = (3𝑥 − 75)°;
𝐵̂ = (165 − 𝑥)° ; 𝐶 = (2𝑥 − 30)°𝑎𝑛𝑑 𝐷̂ = (2𝑥 + 30)° . Find the nature of this quadrilateral.
2. In the quadrilateral ABCD, we have : AB=5𝑥 + 2 , BC= 2𝑥 + 3 , CD= 6𝑥 + 1 and AD=5x.
What is the nature of ABCD if its perimeter is equal to 24 cm.
3. Verify that
1−𝑎
3+2𝑏
is the inverse of
−2𝑏−3
𝑎−1
with 𝑏 ≠ −
3
2
𝑎𝑛𝑑 𝑎 ≠ 1.
4. Write the scientific notation of 𝐴 =
0.23×10³−1.7×10²
0.5×10⁻¹
5. Write the fractional form of B= 0.2 + 0. 3̅

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IV. (4 points)
Consider a triangle ABC isosceles at A such that
BC= 4cm and 𝐴𝐵̂ 𝐶 = 50°. Place the point E
symmetric of B with respect to A and the point F
symmetric of C with respect to A.
1) What is the nature of the quadrilateral BCEF ?
Justify.
2) Calculate the measure of the angle 𝐹𝐸̂ 𝐴 .
3) The parallel at (BE) passing through C and the
parallel at (FC) passing through E intersect in L.
a. What is the nature of AELC ? Justify.
b. Calculate the measure of the angle 𝐴𝐸̂ 𝐶.
c. Deduce the measure of the angle 𝐶𝐸̂ 𝐿.
V. (3 points)
Calculate and simplify if possible.
a) [
2
3
−
1
1+
1
3
] ÷
1+
1+
1
2
1−
1
3
2
3
−1
c)
7
2
−
3
4
2
3
−
1
4
×
7
2
÷
3+
1
2
1
3
+5−
7
2
b)
1
4+
1
7+
1
1+
1
3
d)
1
4
+
3
4
÷ (1 +
1
8
)
VI. (5 points)
ABCD is a rectangle of center O such that AB= 2AD. Let I be the midpoint [AB], E is the
symmetric of O with respect to I. (IO) cuts [CD] at point F.
1) Draw the figure.
2) What is the nature of quadrilateral AOBE ? Justify.
3) What is the nature of quadrilateral EBCO ? Justify.
4) What is the nature of quadrilateral IBCF ? Justify.
5) (DI) cuts (BC) at point G.
a. Prove that the triangle GCD is right isosceles.
b. Prove that (CI) is the perpendicular bisector of [DG].
GOOD WORK.