Rhombus ABCDABCDA, B, C, D has the following vertices: A(-8,11)A(−8,11)A, left parenthesis, minus, 8, comma, 11, right parenthesis B(-1,-13)B(−1,−13)B, left parenthesis, minus, 1, comma, minus, 13, right parenthesis C(14,7)C(14,7)C, left parenthesis, 14, comma, 7, right parenthesis D(7,31)D(7,31)D, left parenthesis, 7, comma, 31, right parenthesis Is rhombus ABCDABCDA, B, C, D a square, and why?

Here is their argument. given the obtuse angle x, we make a quadrilateral abcd with ∠dab = x, and ∠abc = 90◦, and ad = bc. say the perpendicular bisector to dc meets the perpendicular bisector to ab at p. then pa = pb and pc = pd. so the triangles pad and pbc have equal sides and are congruent. thus ∠pad = ∠pbc. but pab is isosceles, hence ∠pab = ∠pba. subtracting, gives x = ∠pad−∠pab = ∠pbc −∠pba = 90◦. this is a preposterous conclusion – just where is the mistake in the "proof" and why does the argument break down there?

Jack lives on a farm and has a large backyard. the area of his rectangular backyard is of a square mile. if the width of the yard is of a mile, what fraction of a mile is the length of the rectangular backyard?

Mr. brettler is hanging up a 5-foot tall rectangular welcome sign in the gym. he has drawn a sketch on a coordinate grid. he has the lower vertices of the sign at (9, –3) and (–6, –3). where would the upper vertices be on the grid?