Construct an isosceles triangle with base 8 cm and altitude 4 cm. Construct another triangle whose sides are 2/3 times the corresponding sides of the isosceles triangle.

Prove that the lengths of tangents drawn from an external point to a circle are equal. 

23.

The ratio of the sums of the first m and first n terms of an A. P. is m₂: n₂. Show that the ratio of its mth and nth terms is (2m−1):(2n−1). 

Speed of a boat in still water is 15 km/h. It goes 30 km upstream and returns back at the same point in 4 hours 30 minutes. Find the speed of the stream. 

Peter throws two different dice together and finds the product of the two numbers obtained. Rina throws a die and squares the number obtained. Who has the better chance to get the number 25. 

In a hospital used water is collected in a cylindrical tank of diameter 2 m and height 5 m. After recycling, this water is used to irrigate a park of the hospital whose length is 25 m and breadth is 20 m. If the tank is filled completely then what will be the height of standing water used for irrigating the park. Write your views on recycling of water. 

In the given figure, the side of square is 28 cm and radius of each circle is half of the length of the side of the square where O and O' are centres of the circles. Find the area of shaded region.

If a≠b≠0, prove that the points (a, a2), (b, b2) (0, 0) will not be collinear.

A chord PQ of a circle of radius 10 cm subtends an angle of 60° at the centre of the circle. Find the area of major and minor segments of the circle. 

The angle of elevation of a cloud from a point 60 m above the surface of the water of a lake is 30° and the angle of depression of its shadow in water of lake is 60°. Find the height of the cloud from the surface of the water.