Properties of lines, rays, line sgement, parallel lines, angles, angle between intersecting lines, polygons, sum of interior angles of polygons, number of diagonals in a polygon, convex and concave polygons. properties of triangles, median, altitude, perpendicular bisector, angle bisector, circles, chords, tangent, quadrilaterals, tangent secant theorem, inscribed circle and circumscribing circles.
Area of triangles, circles, quadrilaterals (including parallelogram, rhombus, rectangle, square, trapezium, and kite) and regular hexagon, surface area and volume of cube, cylinder, cone, sphere and hemisphere, length of the longest diagonal in a cube and cuboid (rectangular cube), volume of water displaced in a vessel when another object is dropped into it, and volume of solids formed by rotation of 2D shapes. A relatively easy topic. Key to success in this topic is to remember all the formulae.
A sector of a circle of radius 5 cm is recast into a right circular cone of height 4 cm. What is the volume of the resulting cone?
Step 1: The radius of the sector will be the slant height of the cone.
Step 2: The height of the cone, slant height of the cone and the radius form a right triangle in which the slant height is the hypotenuse.
Step 3: Using data about slant height and height of the cone, compute radius of cone.
Step 4: Compute volume of cone.
A cylindrical vessel is filled with water up to some height. If a sphere of diameter 8 cm is dropped into the cylinder, the water level rises by half of the initial level. Instead, if a sphere of diameter 16 cm is dropped, the water level rises to a height h_{2}. What percentage of this new height h_{2} is the initial level of water?
Step 1: Assign variables for the initial height of water in the cylindrical vessel and the height to which water rises after sphere 1 is dropped.
Step 2: Compute volume of sphere 1 and equate it to the increase in volume of water in the cylindrical vessel
Step 3: Compute volume of sphere 2 and equate it to the increase in volume of water in the cylindrical vessel
Step 4: Solve equations obtained in steps 2 and 3 to find the relation between h_{2} and the initial level of water in the vessel.
Chord AC at a distance of 7 cm from the center of a circle subtends an angle of 120 degrees at the center. What is the area of major segment?
Step 1: Because the angle subtended by the chord at the center of the circle is 120°, we have a 30 - 60 - 90 right triangle.
Step 2: Using ratio of measure of sides of a 30 - 60 - 90 right triangle, compute hypotenuse of the triangle, which is the radius of the circle, and the base of the triangle.
Step 3: Compute are of minor sector.
Step 4: Compute are of triangle AOC and subtract it from area of minor sector to compute area of minor segment.
Step 5: Area of major segment = Area of circle - area of minor segment
The area for which of the following will necessarily be more than 50 square units.
Indicate all such expressions
Compute the area of each of the six 2D shapes given and mark all those which have an area greater than 50 square units.
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