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#### The points A(1, 7), B(20, 7) and C( p, q) form the vertices of a triangle ABC, as shown in the diagram. The point D(8, 2) is the mid-point of AC.

20 viewed last edited 2 years ago Anonymous
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The line l, which passes through D and is perpendicular to AC, intersects AB at E.

(a) Find an equation for l, in the form ax + by + c = 0, where a, b and c are integers

(b) Find the exact x-coordinate of E.  Krishna
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Step 1: Note down the given points as well as assign these points with the variables

EXAMPLE: ( 7, 4 ) assign this point with (x_1, y_1)

( 2, 0 ) assign this point with (x_2,y_2)

................etc

Step 2: Calculate gradient or slope(m) from two points

FORMULA: \frac{y-change}{x-change},

or \frac{y_2 - y_1}{x_2 - x_1}

EXAMPLE: Slope of A(7, 4) and B(2, 0 )

\frac{4-0}{7-2} = \frac{4}{5}

Step 3: Find the slope of the another line.

NOTE: i) Parallel lines have the same slope.

ii) Perpendicular lines have slopes that are opposite reciprocals, like

\frac{a}{b} \text{ and } \frac{-b}{a}. The slopes also have a product of -1

Step 4: Substitute either values slope(m) and any one point into the equation of a straight line.

FORMULA: Equation of a straight line

y - y_1 = m(x - x_1)

Where m = slope and ( x_1, y_1) = any point

EXAMPLE: I took (2, 0) as a point and m = \frac{4}{5}

y - 0 = \frac{4}{5} ( x - 2)

5y = 4x - 8

Step 4: Simplify and make the equation in the form of Ax + By + C =0

b) Find the exact x-coordinate of E.

Step 1: Take the calculated line equation of the of the perpendicular line

Step 2: Notice the coordinate from the diagram and substitute in the line equation to get the another coordinate.

NOTE: From the diagram AB line parallel to the x-axis at y = 7. Hence, any point on the line has same y coordinator. So substitute y = 7 in the line equation to get the x-coordinator

Step 3: Simplify the equation

Apply the BODMAS rules