Triangles
In a right-angled triangle ABC at A, if AB=6 cm and AC=8 cm, calculate the length of [BC].
10 cm
Two circles (C) and (C′) have radii R=6 cm and r=4 cm. If the distance between their centers is OO′=10 cm, what is their relative position?
Tangent externally
Expand and reduce the expression using a remarkable identity: (2x+3)2.
4x2 + 12x + 9
In a triangle ABC, M is the midpoint of [AB] and N is the midpoint of [AC]. If BC=12 cm, what is the length of segment [MN] and its position relative to (BC)?
MN=6 cm and (MN)∥(BC)
Two circles have radii R=7 cm and r=3 cm. If the distance between their centers is OO′=2 cm, what is their relative position?
Internally Disjoint
Factorize the expression completely by finding the common factor: 3x(x − 5) + 2(x − 5).
(x − 5)(3x + 2)
In a semi-equilateral triangle OPQ right-angled at P, the angle POQ=60. If the hypotenuse OQ=10 cm, what is the length of the side opposite to the 30 angle, [OP]?
OP=5 cm
Two circles (C) and (C′) are tangent internally. The radius of the larger circle is R=9 cm and the distance between their centers is OO' = 4. Find the value of the smaller radius, r.
r=5 cm
Solve the following equation: (2x−4)(3x+9)=0.
x=2 or x=−3
In a triangle ABC, I is the midpoint of [AB]. A line passing through I parallel to (BC) cuts [AC] at J. If AC=14 cm, calculate the length of [AJ]
AJ=7 cm
Two circles (C) and (C′) with radii R=2x+1 and r=x are tangent externally. Knowing that the distance between their centers is OO′=13 cm, find the value of x.
x=4
Consider the two algebraic expressions:M(x)=(3x−2)2 N(x)=(x+4)2
Solve the equation: M(x)−N(x)=0
x=3 or x=−0.5
In a semi-equilateral triangle ABC right-angled at A, the angle ABC=30∘ and ACB=60∘. Knowing that the shortest side is AC=5 cm and the hypotenuse is BC=10 cm, calculate the exact length of the side facing the 60∘ angle, [AB].
5 radical(3) cm
Two circles (C) and (C′) have radii R=x+8 and r=x+2. The distance between their centers is given by the expression OO′=3x−2. If these two circles are tangent internally, find the value of x, and calculate the distance OO′.
x = 4 and OO′=10 cm
Consider the expression: E(x)=(2x−5)2−9.
Solve the equation E(x)=0.
x=4 or x=1