The perpendicular from the square corner of a right-triangle divides the opposite side into two parts of lengths $a$ and $b$. Taking the length of the perpendicular as $h$, prove that $h^2=ab$.

At two ends of a horizontal line, angles of equal size are drawn, and two points on the slanted lines are joined. Prove that the parts of the horizontal line and parts of the slanted line are in the same ratio.

At two ends of a horizontal line, angles of equal size are drawn, and two points on the slanted lines are joined. Prove that the two slanted lines at the ends of the horizontal line are in the same ratio.

At two ends of a horizontal line, angles of equal size are drawn and two points on the slanted lines are joined. Explain how a line of length$6 \mathrm{cm}$ can be divided in the ratio $3 : 4$.

The midpoint of the bottom side of a square is joined to the ends of the top side and extended by the same length. The ends of these lines are joined and perpendiculars are drawn from these points to the bottom side of the square extended. Prove that the quadrilateral obtained thus is also a square.

Explain how we can draw a square with two corners on a semicircle and the other two corners on its diameter as given in the figure.

The picture shows a square drawn sharing one corner with a right triangle and the other three corners on the sides of this triangle. Calculate the length of a side of the square.

The picture shows a square drawn sharing one corner with a right triangle and the other three corners on the sides of this triangle.

What is the length of a side of the square drawn like this within a triangle of sides $3 \mathrm{cm}, 4 \mathrm{cm}$ and $5 \mathrm{cm}$.

Two poles of heights $3 \mathrm{m}$ and $2 \mathrm{m}$ are erected upright on the ground and ropes are stretched from the top of each to the foot of the other.

At what height  above the ground do the ropes cross each other?