Special Lines in Triangles
Construct the midsegment of the triangle below. A midsegment is a line segment that joins the midpoints of two sides of a triangle or trapezoid. For the moment, we will work with a triangle.
a. Use your compass and straightedge to determine the midpoints of 𝐴B and 𝐴C as 𝑋 and 𝑌, respectively.
b. Draw midsegment XY.
Compare ∠AXY and ∠ABC; compare ∠AYX and ∠ACB. Without using a protractor, what would you guess is the relationship between these two pairs of angles? What are the implications of this relationship?
Note that though we chose to determine the midsegment of AB and AC, we could have chosen any two sides to work with. Let us now focus on the properties associated with a midsegment.
The midsegment of a triangle is parallel to the third side of the triangle and half the length of the third side of the triangle.
We can prove these properties to be true. Continue to work with the figure from the Opening Exercise.
Given: XY is a midsegment of △ ABC.
Prove: XY ∥ BC and XY = 1/2 BC
Construct the following: In the Opening Exercise figure, draw △ YGC according to the following steps. Extend XY to point 𝐺 so that YG = XY. Draw GC.
(1) What is the relationship between XY and YG? Explain why.
(2) What is the relationship between ∠AYX and ∠GYC? Explain why.
(3) What is the relationship between AY and YC? Explain why.
(4) What is the relationship between △ AXY and △ CGY? Explain why.
(5) What is the relationship between GC and AX? Explain why.
(6) Since AX = BX, what other conclusion can be drawn? Explain why.
(7) What is the relationship between 𝑚∠AXY and 𝑚∠YGC? Explain why
(8) Based on (7), what other conclusion can be drawn about AB and GC? Explain why.
(9) What conclusion can be drawn about BXGC based on (7) and (8)? Explain why.
(10) Based on (9), what is the relationship between XG and BC?
(11) Since YG = XY, XG = __ XY. Explain why.
(12) This means BC = __ XY. Explain why.
(13) Or by division, XY = __ BC.
Note that Steps (9) and (13) demonstrate our Prove statement
If a quadrilateral is a parallelogram, then its opposite sides and angles are equal in measure. Complete the diagram, and develop an appropriate Given and Prove for this case. Use triangle congruence criteria to demonstrate why opposite sides and angles of a parallelogram are congruent.
Construction: Label the quadrilateral 𝐴𝐵𝐶𝐷, and mark opposite sides as parallel. Draw diagonal 𝐵𝐷.
If a quadrilateral is a parallelogram, then the diagonals bisect each other. Complete the diagram, and develop an appropriate Given and Prove for this case. Use triangle congruence criteria to demonstrate why diagonals of a parallelogram bisect each other. Remember, now that we have proved opposite sides and angles of a parallelogram to be congruent, we are free to use these facts as needed (i.e., 𝐴𝐷 = 𝐶𝐵, 𝐴𝐵 = 𝐶𝐷, ∠𝐴 ≅ ∠𝐶, ∠𝐵 ≅ ∠𝐷).
Construction: Label the quadrilateral 𝐴𝐵𝐶𝐷. Mark opposite sides as parallel. Draw diagonals 𝐴𝐶 and 𝐵𝐷.
Now we have established why the properties of parallelograms that we have assumed to be true are in fact true. By extension, these facts hold for any type of parallelogram, including rectangles, squares, and rhombuses. Let us look at one last fact concerning rectangles. We established that the diagonals of general parallelograms bisect each other. Let us now demonstrate that a rectangle has congruent diagonals.
Apply what you know about the properties of midsegments to solve the following exercises.
Perimeter of △ ABC =
In △ RST, the midpoints of each side have been marked by points 𝑋, 𝑌, and 𝑍.
a. What conclusion can you draw about the four triangles within △ RST? Explain why.
b. State the appropriate correspondences among the four triangles within △ RST.
c. State a correspondence between △ RST and any one of the four small triangles
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