Proofs with Variable Coordinates Page 13: #’s 17-21

17.The vertices of quadrilateral RSTV are R(0,0), S(a,0), T(a+b,c) and V(b,c) a) Find the slopes of RV and ST

17.The vertices of quadrilateral RSTV are R(0,0), S(a,0), T(a+b,c) and V(b,c) b) Find the lengths of RV and ST c) Since one pair of opposite sides has equal slopes, they are parallel. The same pair of opposite sides are equal in length. Quadrilateral RSTV with one pair of opposite sides both parallel and congruent is a parallelogram.

18. The vertices of triangle ABC are A(0,0), B(4a,0), C(2a,2b) a) Find the coordinates of D, the midpoint of AC.

18. The vertices of triangle ABC are A(0,0), B(4a,0), C(2a,2b) b) Find the coordinates of E, the midpoint of BC.

18. The vertices of triangle ABC are A(0,0), B(4a,0), C(2a,2b) c) Show that AB=2DE ?

19. The vertices of quadrilateral ABCD are A(0,0), B(a,0), C(a,b) and D(0,b) a) Show that ABCD is a parallelogram Since the slopes of both pairs of opposite sides are equal, they are parallel. Therefore quadrilateral ABCD with both pair of opposite sides parallel is a parallelogram.

19. The vertices of quadrilateral ABCD are A(0,0), B(a,0), C(a,b) and D(0,b) b) Show that diagonal AC is congruent to diagonal BD C) The diagonals of parallelogram ABCD are congruent. A parallelogram with congruent diagonals is a rectangle.

20. The vertices of quadrilateral ABCD are A(0,0), B(r,s), C(r,s+t) and D(0,t) a) Represent the slopes of AB and CD Since the slopes of the opposite sides are equal, they are parallel.

20. The vertices of quadrilateral ABCD are A(0,0), B(r,s), C(r,s+t) and D(0,t) b) Represent the lengths of AB and CD C) Since quadrilateral ABCD has the same pair of opposite sides (AB and CD) both parallel and congruent, it is a parallelogram.

21. The vertices of triangle RST are R(0,0), S(2a,2b), T(4a,0) The midpoints of RS, ST, TR are L, M, and N, respectively. a) Express the coordinates of the midpoints in terms of a and b.

21. The vertices of triangle RST are R(0,0), S(2a,2b), T(4a,0) The midpoints of RS, ST, TR are L, M, and N, respectively. Since the slopes of LM and RT are equal, they are parallel.

21. The vertices of triangle RST are R(0,0), S(2a,2b), T(4a,0) The midpoints of RS, ST, TR are L, M, and N, respectively. Since SN has an undefined slope, it is a vertical line. RT has a zero slope so it is a horizontal line. Therefore SN is perpendicular to RT.

21. The vertices of triangle RST are R(0,0), S(2a,2b), T(4a,0) The midpoints of RS, ST, TR are L, M, and N, respectively. Since two sides of the triangle are congruent, triangle RST is isosceles.