Prove that the area of a rhombus is one-half the product of the lengths of the diagonals. Suppose that Delta ABCis equilateral triangle and that P is a point in the interior of this triangle. Prove that the sum of the perpendicular distances from P to each of the sides of the triangle is equal to the height of the triangle. Prove that the area of a triangle is equal to one-half the product of its perimeter and the length of the radius of a circle inscribed within the triangle. Prove that the diagonal of a kite connecting the vertices where the congruent sides intersect bisects the angles at these vertices and is the perpendicular bisector of the other diagonal. (Use the strict definition of a kite which is a quadrilateral with two distinct pairs of congruent adjacent sides.)

1. Prove that the area of a rhombus is one-half the product of the lengths of the diagonals.
2. Suppose that $\mathrm{\Delta }ABC$ is equilateral triangle and that P is a point in the interior of this triangle. Prove that the sum of the perpendicular distances from P to each of the sides of the triangle is equal to the height of the triangle.
3. Prove that the area of a triangle is equal to one-half the product of its perimeter and the length of the radius of a circle inscribed within the triangle.
4. Prove that the diagonal of a kite connecting the vertices where the congruent sides intersect bisects the angles at these vertices and is the perpendicular bisector of the other diagonal. (Use the strict definition of a kite which is a quadrilateral with two distinct pairs of congruent adjacent sides.)
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Given ABCD is a rhombus the diagonal AC and BD cut at point O
Then $\angle AOD=\angle AOB=\angle COD=\angle BOC=900$
The area of rhombus ABCD divided diagonal in four parts
So area of rhombus ABCD =area of triangle AOD+area of triangle AOB+area of triangle BOC+area of triangle COD
=$21×AO×OD+21×AO×OB+21×BO×OC+21×OD×OC$
=$21×AO\left(OD+OB\right)+21OC\left(BO+OD\right)$
=$21×AO×BD+21×OC×BD$
=$21BD\left(AO+OC\right)=21×BD×AC$
So area of rhombus is equal to half of the product of diagonals....