19. Proof Flag Design The flag design below is made by connecting the midpoints of the sides of a rectangle. Use coordinate geometry to prove that the quadrilateral formed is a rhombus.

    

20. Open-Ended Give an example of a statement that you think is easier to prove with a coordinate geometry proof than with a proof method that does not require coordinate geometry. Explain your choice.

Use coordinate geometry to prove each statement.

21. Proof Think About a Plan If a parallelogram is a rhombus, its diagonals are perpendicular (Theorem 6-13).
    • How will you place the rhombus in a coordinate plane?
    • What formulas will you need to use?
22. The altitude to the base of an isosceles triangle bisects the base.
23. If the midpoints of a trapezoid are joined to form a quadrilateral, then the quadrilateral is a parallelogram.
24. One diagonal of a kite divides the kite into two congruent triangles.
25. Proof You learned in Theorem 5-8 that the centroid of a triangle is two thirds the distance from each vertex to the midpoint of the opposite side. Complete the steps to prove this theorem.
    1. Find the coordinates of points L, M, and N, the midpoints of the sides of Δ A B C . cap delta eh b c .
    2. Find equations of A M ↔ , B N ↔ , modified eh m with left right arrow above . comma , modified b n with left right arrow above , comma  and C L ↔ . modified c l with left right arrow above , .
    3. Find the coordinates of point P, the intersection of A M ↔ modified eh m with left right arrow above  and B N ↔ . modified b n with left right arrow above , .
    4. Show that point P is on C L ↔ . modified c l with left right arrow above , .

    5. Use the Distance Formula to show that point P is two thirds the distance from each vertex to the midpoint of the opposite side.

       

26. Proof Complete the steps to prove Theorem 5-9. You are given Δ A B C cap delta eh b c  with altitudes p, q, and r. Show that p, q, and r intersect at a point (called the orthocenter of the triangle).

    

    1. The slope of B C ¯ b c bar  is c − b . c over negative b , .  What is the slope of line p?
    2. Show that the equation of line p is y = b c ( x − a ) . y equals , b over c , open x minus eh close .
    3. What is the equation of line q?
    4. Show that lines p and q intersect at ( 0 , − a b c ) . open 0 comma . fraction negative eh b , over c end fraction . close .
    5. The slope of A C ¯ eh c bar  is c − a . c over negative eh , .  What is the slope of line r?
    6. Show that the equation of line r is y = a c ( x − b ) . y equals , eh over c , open x minus b close .
    7. Show that lines r and q intersect at ( 0 , − a b c ) . open 0 comma . fraction negative eh b , over c end fraction . close .
    8. What are the coordinates of the orthocenter of Δ A B C ? cap delta eh b c question mark

C Challenge

27. Multiple Representations Use the diagram below.

    
    Image Long Description

    1. Explain using area why 1 2 a d = 1 2 b c 1 half , eh d equals , 1 half , b c  and therefore a d = b c . eh d equals b c .
    2. Find two ratios for the slope of ℓ . script l .  Use these two ratios to show that a d = b c . eh d equals b c .

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