Given: The coordinates of triangle PQR are P(0, 0), Q(2a, 0), and R(2b, 2c). Prove: The line containing the midpoints of two sides of

Question

Given: The coordinates of triangle PQR are P(0, 0), Q(2a, 0), and R(2b, 2c).
Prove: The line containing the midpoints of two sides of a triangle is parallel to the third side.

As part of the proof, find the midpoint of PR

in progress 0
Kennedy 2 weeks 2021-09-13T11:57:39+00:00 2 Answers 0

Answers ( )

    0
    2021-09-13T11:59:13+00:00

    Answer:

    b,c

    Step-by-step explanation:

    That guy above took so long

    0
    2021-09-13T11:59:21+00:00

    Answer:

    The line containing the midpoints of two sides of a triangle is parallel to the third side ⇒ proved down

    Step-by-step explanation:

    * Lets revise the rules of the midpoint and the slope to prove the

      problem

    – The slope of a line whose endpoints are (x1 , y1) and (x2 , y2) is

      m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}

    – The mid-point of a line whose endpoints are (x1 , y1) and (x2 , y2) is

      (\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})

    * Lets solve the problem

    – PQR is a triangle of vertices P (0 , 0) , Q (2a , 0) , R (2b , 2c)

    – Lets find the mid-poits of PQ called A

    ∵ Point P is (x1 , y1) and point Q is (x2 , y2)

    ∴ x1 = 0 , x2 = 2a and y1 = 0 , y2 = 0

    ∵ A is the mid-point of PQ

    A=(\frac{0+2a}{2},\frac{0+0}{2})=(\frac{2a}{2},\frac{0}{2})=(a,0)

    – Lets find the mid-poits of PR which called B

    ∵ Point P is (x1 , y1) and point R is (x2 , y2)

    ∴ x1 = 0 , x2 = 2b and y1 = 0 , y2 = 2c

    ∵ B is the mid-point of PR

    B=(\frac{0+2b}{2},\frac{0+2c}{2})=(\frac{2b}{2},\frac{2c}{2})=(b,c)

    – The parallel line have equal slopes, so lets find the slopes of AB and

      QR to prove that they have same slopes then they are parallel

    # Slope of AB

    ∵ Point A is (x1 , y1) and point B is (x2 , y2)

    ∵ Point A = (a , 0) and point B = (b , c)

    ∴ x1 = a , x2 = b and y1 = 0 and y2 = c

    ∴ The slope of AB is m=\frac{c-0}{b-a}=\frac{c}{b-a}

    # Slope of QR

    ∵ Point Q is (x1 , y1) and point R is (x2 , y2)

    ∵ Point Q = (2a , 0) and point R = (2b , 2c)

    ∴ x1 = 2a , x2 = 2b and y1 = 0 and y2 = 2c

    ∴ The slope of AB is m=\frac{2c-0}{2b-2a}=\frac{2c}{2(b-c)}=\frac{c}{b-a}

    ∵ The slopes of AB and QR are equal

    ∴ AB // QR

    ∵ AB is the line containing the midpoints of PQ and PR of Δ PQR

    ∵ QR is the third side of the triangle

    ∴ The line containing the midpoints of two sides of a triangle is parallel

      to the third side

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