Points A(−8,2), B(−2,−6) and D(0,8) are vertices of a triangle that lie on the circumference of a circle. The…

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Points A(−8,2), B(−2,−6) and D(0,8) are vertices of a triangle that lie on the circumference of a circle. The centre of the circle is M and BD is a diameter of the circle. Calculate the coordinates of M. Find the equation of the line parallel to BD and passing through the point A.

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(a) Calculate the coordinates of M. The problem states that BD is a diameter of the circle and M is the center of the circle. Therefore, M is the midpoint of the line segment BD. Step 1: Identify the coordinates of points B and D. B = (−2, −6); D = (0, 8); Step 2: Use the midpoint formula M = ((x₁+x₂)/(2), (y₁+y₂)/(2)). M = ((−2+0)/(2), (−6+8)/(2)) Step 3: Calculate the coordinates of M. M = ((−2)/(2), 2/2); M = (−1, 1) The coordinates of M are M(−1, 1) (b) Find the equation of the line parallel to BD and passing through the point A. Step 1: Calculate the gradient (slope) of the line BD. The coordinates are B(−2, −6) and D(0, 8). The gradient m = (y₂ − y₁)/(x₂ − x₁). mBD = (8 − (−6))/(0 − (−2)); mBD = (8 + 6)/(0 + 2); mBD = 14/2; mBD = 7 Step 2: Determine the gradient of the line parallel to BD. Since the required line is parallel to BD, it will have the same gradient. So, mparallel = 7. Step 3: Use the point-slope form of a linear equation, y − y₁ = m(x − x₁), with point A(−8, 2) and mparallel = 7. y − 2 = 7(x − (−8)); y − 2 = 7(x + 8); Step 4: Simplify the equation into the form y = mx + c. y − 2 = 7x + 56; y = 7x + 56 + 2; y = 7x + 58; The equation of the line is y = 7x + 58

Points A(-8,2), B(-2,-6) and D(0,8) are vertices of a triangle that lie on the circumference of a circle. The centre of the circle is M and BD is a diameter of the circle. Calculate the coordinates of M. Find the equation of the line parallel to BD and passing through the point A.

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Step-by-step solution

Part a· 3 steps

Identify the coordinates of points $B$ and $D$ .

Use the midpoint formula $M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$ .

Calculate the coordinates of $M$ .

Part b· 4 steps
$\text{}y = 7x + 58\text{}$

Calculate the gradient (slope) of the line $BD$ .

Determine the gradient of the line parallel to $BD$ .

Use the point-slope form of a linear equation, $y - y_1 = m(x - x_1)$ , with point $A(-8, 2)$ and $m_{parallel} = 7$ .

Simplify the equation into the form $y = mx + c$ .

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Points A(-8,2), B(-2,-6) and D(0,8) are vertices of a triangle that lie on the circumference of a circle. The centre of the circle is M and BD is a diameter of the circle. Calculate the coordinates of M. Find the equation of the line parallel to BD and passing through the point A.

Answers

Step-by-step solution

Part a· 3 steps

Identify the coordinates of points BBB and DDD.

Use the midpoint formula M=(x1+x22,y1+y22)M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)M=(2x1​+x2​​,2y1​+y2​​).

Calculate the coordinates of MMM.

Part b· 4 stepsy=7x+58\text{}y = 7x + 58\text{}y=7x+58

Calculate the gradient (slope) of the line BDBDBD.

Determine the gradient of the line parallel to BDBDBD.

Use the point-slope form of a linear equation, y−y1=m(x−x1)y - y_1 = m(x - x_1)y−y1​=m(x−x1​), with point A(−8,2)A(-8, 2)A(−8,2) and mparallel=7m_{parallel} = 7mparallel​=7.

Simplify the equation into the form y=mx+cy = mx + cy=mx+c.