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
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
