In the following diagram, the point R is the center of the circle. The lines PQ and ZV are tangential to the circle. The relation among the areas of the squares, PXWR, RUVZ and SPQT is
To solve this problem, let's analyze the given diagram and understand the relationship between the squares and the circle.
The point R is the center of the circle. The lines PQ and ZV are tangent to the circle, meaning they meet the circle at one point each.
- Since PQ and ZV are tangential, the lines perpendicular to the radius at the points of tangency will match the sides of the squares.
- Let's denote the side of square SPQT as \(a\), the side of square PXWR as \(b\), and the side of square RUVZ as \(c\).
- The length of line PR is the radius of the circle plus the side of square SPQT, and similarly, RZ is also the radius plus the side of square RUVZ.
- The relation between these areas is given by the geometry of the tangents. Each tangent forms a right-angled triangle with the radius of the circle and the side of the square:
- The area of square PXWR is \(b^2\).
- The area of square RUVZ is \(c^2\).
- The area of square SPQT is \(a^2\).
- Since PXWR = SPQT + RUVZ, we have the equation:
- \(a^2 = b^2 - c^2\)
This matches option two:
- Area of SPQT = Area of PXWR – Area of RUVZ
Thus, the correct answer is: Area of SPQT = Area of PXWR – Area of RUVZ.
Important Questions from Mensuration and Geometry
The city of Atlantis was crafted by the God of the seas, Poseidon. It was made of alternating concentric circular rings of land (shaded) and water (not shaded) as represented in the figure (not to scale). The radius of Inner Island was 2.5 stades (a unit of length used in ancient Greece). The water surrounding Inner Island was one stade wide (length AB). This was surrounded by two pairs of alternating rings of land and water. The first pair of land and water was two stades wide each (lengths BC and CD), and the outer pair is three stades wide each (lengths DE and EF).
The ratio of the surface area of the land to that of the water in the city of Atlantis is _________ (round off to two decimal places).
In the given figure, $P, Q$, and $R$ are three points on a circle of radius 10 cm with $O$ as its center, $\overline{PQ} = \overline{RQ}$, and $\angle PQR = 45^\circ$. The figure is representative.
The area of the shaded region $PQRO$ is ______________ cm$^2$.
- A straight line $y = x - 1$ intersects a circle with center at $x = 1, y = 1$ and radius of magnitude 1 at two points. The length of the chord formed by this intersection is _______. (rounded off to three decimal places)
- The shell of a hollow spherical nanoparticle has a uniform thickness of 3 nanometers (nm). The outer radius of the nanoparticle is 5 nm. The ratio of the volume of the shell to the volume of the hollow core is ________
(Round off to one decimal place) - The volume of a sphere of diameter 1 unit is ______ than the volume of a cube of side 1 unit.
