Let ABCD be a general quadrilateral and let the points P, Q,R, and S be the midp

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Let ABCD be a general quadrilateral and let the points P, Q,R, and S be the midpoints of AB bar, BC bar, CD bar, and DA bar, respectively. Prove that PQRS is a parallelogram. triangleABD similarity triangleAPS, triangle BDC similarity triangle QRS, triangle ACD similarity triangle SRD, and triangle ABC similarity triangle PBQ. Then, use those to show that the opposite sides of PQRS must be congruent Let ABCDE and FGHIJ be (not necessarily regular) pentagons that are similar to each other with scale factor sigma. Prove that the ratio of the areas of these two pentagons is sigma^2. From now on, we can assume that the ratio of the areas of any two similar polygons is the square of the scale factor. Let P_1, and P_2 be similar polygons with scale factor sigma. If P_1 has perimeter 6,000 inches and area 4,000 square inches, and P_2 has area 10 square inches, find the following: the value of the scale factor, sigma the perimeter of P_2 Construct a counter example to the following statement: If the diagonals of rectangle ABCD are congruent to the diagonals of rectangle EFGH, then ABCD congruent EFGH. Make sure you justify the accuracy of your proposed counterexample. Let triangle ABC be a triangle, and let m triangle BAC = theta. Prove that A_ABC = 1/2 (AB){AC) sin theta.

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