If the area of triangle abc whose vertices are a(4, k) , b ( -2, -4 ) and c ( 6, -3) is 66 sqaure units ,find the 'k'

Question

Answer

Answer

Answer

Given Information:
The vertices of triangle ABC are A(4, k), B(-2, -4), and C(6, -3).
The area of triangle ABC is 66 square units.

To Find:
The value of k.

Approach:
To find the value of k, we need to calculate the area of triangle ABC using the given coordinates of its vertices and equate it to 66 square units. We can then solve the resulting equation to find the value of k.

Solution:

Step 1: Calculate the length of the base of the triangle.
To find the length of the base, we can use the coordinates of points B and C.
Length of base = Distance between points B and C
Using the distance formula, the length of base BC can be calculated as follows:
BC = √((x2 - x1)^2 + (y2 - y1)^2) [Where (x1, y1) = B(-2, -4) and (x2, y2) = C(6, -3)]

BC = √((6 - (-2))^2 + (-3 - (-4))^2)
BC = √((6 + 2)^2 + (-3 + 4)^2)
BC = √(8^2 + 1^2)
BC = √(64 + 1)
BC = √65

Step 2: Calculate the height of the triangle.
The height of the triangle can be calculated as the perpendicular distance from point A to the line containing base BC.
To find the equation of the line containing BC, we can use the slope-intercept form.
Slope of BC = (y2 - y1) / (x2 - x1)
Slope of BC = (-3 - (-4)) / (6 - (-2))
Slope of BC = (-3 + 4) / (6 + 2)
Slope of BC = 1 / 8

Using the point-slope form of a line, we can write the equation of the line containing BC as:
y - y1 = m(x - x1) [Where (x1, y1) = C(6, -3) and m is the slope of BC]

y - (-3) = (1/8)(x - 6)
y + 3 = (1/8)(x - 6)
8(y + 3) = x - 6
8y + 24 = x - 6
x - 8y - 30 = 0

The equation of the line containing BC is x - 8y - 30 = 0.

Now, the height of the triangle is the perpendicular distance from point A(4, k) to the line x - 8y - 30 = 0.
The formula to calculate the perpendicular distance from a point to a line ax + by + c = 0 is given by:
Distance = |ax1 + by1 + c| / √(a^2 + b^2) [Where (x1, y1) is the point]

Substituting the values, we have:
Distance = |(1)(4) + (-8)(k) + (-30)| / √(1^

Answer

Use area of triangle =66

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