Section 6-7

Proofs Using Coordinate Geometry

 

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What You'll Learn

  1. To prove theorems using figures in the coordinate plane

… And Why

To use coordinate geometry to prove that a flag design includes a rhombus, as in Example 2

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Building Proofs in the Coordinate Plane

In Lesson 5-1, you learned about midsegments of triangles. A trapezoid also has a midsegment. The midsegment of a trapezoid is the segment that joins the midpoints of the nonparallel opposite sides. It has two unique properties.

 

Key Concepts

Theorem 6-18  Trapezoid Midsegment Theorem

Image shows a trapezoid T R A P, with midsegment M N parallel to bases R A and P T. R M and M T are congruent. A N and N P are congruent.(1) The midsegment of a trapezoid is parallel to the bases.
(2) The length of the midsegment of a trapezoid is half the sum of the lengths of the bases.

segment M N. || segment T P., segment M N. || segment R A., and MN = one half(TP + RA).

Formulas for slope, midpoint, and distance are used in a proof of Theorem 6-18.

arrow indicating example contains a proof Example 1 Planning a Coordinate Geometry Proof

Developing Proof Plan a coordinate proof of Theorem 6-18.

  1. Given  segment M N. is the midsegment of trapezoid TRAP.
  2. Prove segment M N. || segment T P., segment M N. || segment R A., and MN = one half.(TP + RA).
  3. Plan Place the trapezoid in the coordinate plane with a vertex at the origin and a base along the x-axis. Since midpoints will be involved, use multiples of 2 to name coordinates. To show lines are parallel, check for equal slopes. To compare lengths, use the Distance Formula.

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Image of a flag. The midpoints of the sides are connected, and the flag is colored blue inside the lines connecting the midpoints. 

Example 2 Real-World globe Connection

The rectangular flag at the left is constructed by connecting the midpoints of its sides. Use coordinate geometry to prove that the quadrilateral formed by connecting the midpoints of the sides of a rectangle is a rhombus.

  1. Given  MNPO is a rectangle.
    T, W, V, U are midpoints of its sides.
  2. Prove TWVU is a rhombus.
  3. Plan Place the rectangle in the coordinate plane with two sides along the axes.
    Use multiples of 2 to name coordinates.
    A rhombus is a parallelogram with four congruent sides. From Lesson 6-6, Example 2, you know that TWVU is a parallelogram. To show segment T W. approximately equal to Isegment W V. approximately equal to segment V U. approximately equal to segment U T., use the Distance Formula.

Coordinate Proof: By the Midpoint Formula, the coordinates of the midpoints are T(0, b), W(a, 2b), V(2a, b), and U(a, 0). By the Distance Formula,

TW = Image shows the distance formula for T W, which is the square root of the entire quantity (a minus 0) squared + (2 b minus b) squared. = the square root of the quantity a squared plus b squared.

WV = Image shows the distance formula for W V, which is the square root of the entire quantity (2 a minus a) squared + (b minus 2 b) squared. = the square root of the quantity a squared plus b squared.

VU = Image shows the distance formula for V U, which is the square root of the entire quantity (a minus 2a) squared + (0 minus b) squared. = the square root of the quantity a squared plus b squared.

UT = Image shows the distance formula for U T, which is the square root of the entire quantity (0 minus a) squared + (b minus 0) squared. = the square root of the quantity a squared plus b squared.

segment T W. approximately equal to segment W V. approximately equal to segment V U. approximately equal to segment U T., so parallelogram TWVU is a rhombus.


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