Positional Problems

1003030307

Level: 
B
Let \( k_1 \) and \( k_2 \) be circles with the centres \( S_1 \), \( S_2 \), and the radii of lengths \( 5\,\mathrm{cm} \) and \( 1\,\mathrm{cm} \) consecutively. Given \( |S_1S_2|=5\,\mathrm{cm} \), what is the mutual position of these circles?
The circles \( k_1 \) and \( k_2 \) intersect.
The circles \( k_1 \) and \( k_2 \) are internally tangent.
The circles \( k_1 \) and \( k_2 \) are externally tangent.
The circle \( k_1 \) lies inside the circle \( k_2 \).
The circle \( k_2 \) lies inside the circle \( k_1 \).

1003030306

Level: 
B
Let \( k_1 \) and \( k_2 \) be circles with the centres \( S_1 \), \( S_2 \), and the radii of lengths \( 5\,\mathrm{cm} \) and \( 1\,\mathrm{cm} \) consecutively. Given \( |S_1S_2|=4\,\mathrm{cm}\), what is the mutual position of these circles?
The circles \( k_1 \) and \( k_2 \) are internally tangent.
The circles \( k_1 \) and \( k_2 \) are externally tangent.
The circle \( k_1 \) lies inside the circle \( k_2 \).
The circle \( k_2 \) lies inside the circle \( k_1 \).

1003030305

Level: 
B
Let \( k_1 \) and \( k_2 \) be circles with the centres \( S_1 \), \( S_2 \), and the radii of lengths \( 5\,\mathrm{cm} \) and \( 1\,\mathrm{cm} \) consecutively. Given \( |S_1S_2|=6\,\mathrm{cm}\), what is the mutual position of these circles?
The circles \( k_1 \) and \( k_2 \) are externally tangent.
The circles \( k_1 \) and \( k_2 \) are internally tangent.
The circle \( k_1 \) lies inside the circle \( k_2 \).
The circle \( k_2 \) lies inside the circle \( k_1 \).

1003030304

Level: 
B
Let \( k_1 \) and \( k_2 \) be circles with the centres \( S_1 \), \( S_2 \), and the radii of lengths \( 5\,\mathrm{cm} \) and \( 1\,\mathrm{cm} \) consecutively. Given \( |S_1 S_2 |=7\,\mathrm{cm} \), what is the mutual position of these circles?
The circle \( k_2 \) lies outside the circle \( k_1 \).
The circle \( k_2 \) lies inside the circle \( k_1 \).
The circles \( k_1 \) and \( k_2 \) intersect.
The circles \( k_1 \) and \( k_2 \) are externally tangent.
The circles \( k_1 \) and \( k_2 \) are internally tangent.

1003030303

Level: 
B
Let \( k_1 \) and \( k_2 \) be circles with the centres \( S_1\), \( S_2 \), and the radii of lengths \( 5\,\mathrm{cm} \) and \( 1\,\mathrm{cm} \) consecutively. Given \( |S_1 S_2|=3\,\mathrm{cm} \), what is the mutual position of these circles?
The circle \( k_2 \) lies inside the circle \( k_1 \).
The circle \( k_1 \) lies inside the circle \( k_2 \).
The circles \( k_1 \) and \( k_2 \) intersect.
The circles \( k_1 \) and \( k_2 \) are externally tangent.
The circles \( k_1 \) a \( k_2 \) are internally tangent.

1003030302

Level: 
B
Let \( p \) be a line and \( k \) be a circle with the centre \( S \) and the radius of \( 3\,\mathrm{cm} \). Given the distance of \( p \) and \( S \) is \( 2.8\,\mathrm{cm} \), the line \( p \) is:
a secant of the circle \( k \)
a tangent of the circle \( k \)
an outer line of the circle \( k \)
a chord of the circle \( k \)

1003030301

Level: 
B
Let \( p \) be a line and \( k \) be a circle with the centre \( S \) and the radius of \( 3\,\mathrm{cm} \). Given the distance of \( p \) and \( S \) is \( 4\,\mathrm{cm} \), the line \( p \) is:
an outer line of the circle \( k \)
a secant of the circle \( k \)
a tangent of the circle \( k \)
a chord of the circle \( k \)

9000121708

Level: 
A
Consider a square \(ABCD\) and a point \(E\) on the side \(BC\) such that the angle \( BAE\) has measure \(20^{\circ }\). The point \(F\) is on the side \(CD\) and the length of \(AF\) equals to the length of \(AE\) (i.e. the triangle \(AEF\) is isosceles with \(AF\) and \(AE\) of equal length). Find the measure of the angle \( AEF\).
\(65^{\circ }\)
\(45^{\circ }\)
\(50^{\circ }\)
\(70^{\circ }\)