Circles $C_1$, $C_2$, and $C_3$, with centers $O_1$, $O_2$, and $O_3$ , and radii $r_1$ , $r_2$, and $r_3$, respectively, touch each other as shown in the following figure. Given $r_1 = 2\text{ cm}$, $r_2 = 1\text{ cm}$ and the angle $\angle O_1O_3O_2$ is $90^\circ$, $r_3 =$ _____ cm.
To find the radius \( r_3 \) of the circle \( C_3 \), we need to use the geometry of the given figure, including the fact that \(\angle O_1O_3O_2 = 90^\circ\).
Given: \(r_1 = 2 \text{ cm}\), \(r_2 = 1 \text{ cm}\), \(\angle O_1O_3O_2 = 90^\circ\).
Using the Pythagorean theorem in \(\triangle O_1O_3O_2\):
\(O_1O_2 = O_1O_3 + O_3O_2\\)
The distance between centers \(O_1\) and \(O_2\) is the sum of the radii of the tangent circles:
\(O_1O_2 = r_1 + r_2 = 2 + 1 = 3 \text{ cm}\)
By the geometry of the circle: \(O_1O_3 = r_1 - r_3\\) and \(O_2O_3 = r_2 - r_3\\)
Substituting into the Pythagorean theorem:
\((r_1 - r_3)^2 + (r_2 - r_3)^2 = (r_1 + r_2)^2\)
\( (2 - r_3)^2 + (1 - r_3)^2 = 3^2 \)
Expanding:
\( (4 - 4r_3 + r_3^2) + (1 - 2r_3 + r_3^2) = 9 \)
Combine like terms: \(2r_3^2 - 6r_3 + 5 = 9\)
Simplify: \(2r_3^2 - 6r_3 - 4 = 0\)
Divide throughout by 2: \(r_3^2 - 3r_3 - 2 = 0\)
Solve using the quadratic formula, \(r_3 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -3\), \(c = -2\).
Calculate the discriminant: \(\sqrt{(-3)^2 - 4 \cdot 1 \cdot (-2)} = \sqrt{9 + 8} = \sqrt{17}\)
Therefore, \(r_3 = \frac{3 \pm \sqrt{17}}{2}\)
Selecting the positive value (as the radius cannot be negative), \(r_3 = \frac{3 + \sqrt{17}}{2}\)
The correct answer is: \(\frac{1}{2}(-3 + \sqrt{17} )\)
Important Questions from Mensuration and Geometry
In the given figure, PQRS is a square of side 2 cm and PLMN is a rectangle. The corner L of the rectangle is on the side QR. Side MN of the rectangle passes through the corner S of the square.
What is the area (in cm²) of the rectangle PLMN?
Note: The figure shown is representative.
A regular dodecagon (12-sided regular polygon) is inscribed in a circle of radius $r$ cm as shown in the figure. The side of the dodecagon is $d$ cm. All the triangles (numbered 1 to 12) in the figure are used to form squares of side $r$ cm and each numbered triangle is used only once to form a square.
The number of squares that can be formed and the number of triangles required to form each square, respectively, are:
Note: The figure shown is representative.
- Which one of the following options has the correct sequence of objects arranged in the increasing number of mirror lines (lines of symmetry)?
- A circle with center at $(x, y) = (0.5, 0)$ and radius $= 0.5$ intersects with another circle with center at $(x, y) = (1, 1)$ and radius $= 1$ at two points. One of the points of intersection $(x, y)$ is:
- During a half-moon phase, the Earth-Moon-Sun form a right triangle. If the Moon-Earth-Sun angle at this half-moon phase is measured to be $89.85^{\circ}$, the ratio of the Earth-Sun and Earth-Moon distances is closest to
