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P(-1; 2), R(3;0) and Q(-3; a) are the vertices of a triangle in the Cartesian plane and PR = PQ

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2 parts
1.1
(1, 1)
1.2
Shown:

a=−2a = -2 if the midpoint of QR is (0;−1)(0; -1)

Step-by-step solution

11 steps

Part 1.1· 3 steps

Determine the coordinates of E, the midpoint of PR.

  1. Use the midpoint formula E=(x1+x22,y1+y22)E = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right).

    The coordinates of P are (−1,2)(-1, 2) and R are (3,0)(3, 0).

  2. Substitute the coordinates into the formula.

    E=(−1+32,2+02)E = \left(\frac{-1 + 3}{2}, \frac{2 + 0}{2}\right)

  3. Simplify the expression.

    E=(22,22)E = \left(\frac{2}{2}, \frac{2}{2}\right) E=(1,1)E = (1, 1)

    The coordinates of E are (1,1)\boxed{(1, 1)}.

Part 1.2

a=−2a = -2 if the midpoint of QR is (0;−1)(0; -1)

Show that a=−2a = -2 if the midpoint of QR is (0;−1)(0; -1).

  1. Use the midpoint formula for Q and R.

    The coordinates of Q are (−3,a)(-3, a) and R are (3,0)(3, 0). The midpoint is given as (0,−1)(0, -1).

    MQR=(−3+32,a+02)M_{QR} = \left(\frac{-3 + 3}{2}, \frac{a + 0}{2}\right)

  2. Simplify the expression for the midpoint.

    MQR=(02,a2)M_{QR} = \left(\frac{0}{2}, \frac{a}{2}\right) MQR=(0,a2)M_{QR} = \left(0, \frac{a}{2}\right)

  3. Equate the y-coordinate of the calculated midpoint to the given y-coordinate.

    a2=−1\frac{a}{2} = -1

  4. Solve for aa.

    a=−1×2a = -1 \times 2 a=−2a = -2

    Therefore, it is shown that a=−2\boxed{a = -2}.

Part 1.3· 4 steps

Determine the gradient of EQ.

  1. Identify the coordinates of E and Q.

    From 1.1, E is (1,1)(1, 1). From 1.2, Q is (−3,−2)(-3, -2) (since a=−2a = -2).

  2. Use the gradient formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.

    Let (x1,y1)=(1,1)(x_1, y_1) = (1, 1) and (x2,y2)=(−3,−2)(x_2, y_2) = (-3, -2).

  3. Substitute the coordinates into the formula.

    mEQ=−2−1−3−1m_{EQ} = \frac{-2 - 1}{-3 - 1}

  4. Simplify the expression.

    mEQ=−3−4m_{EQ} = \frac{-3}{-4} mEQ=34m_{EQ} = \frac{3}{4}

    The gradient of EQ is ∗34∗\boxed*{\frac{3}{4}*}.

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