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ArbitrLikvidat [17]
3 years ago
14

Co-ordinate Geometry

Mathematics
1 answer:
saul85 [17]3 years ago
7 0

Answer:

The value of a  = 14

Step-by-step explanation:

Given

(x₁, y₁) = (2, 4)

(x₂, y₂) = (6, a)

(x₃, y₃) = (-1, 1)

A = 9 sq.units

Area of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is:

A=\frac{\left|x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right)\right|}{2}

substituting the values (x₁, y₁) = (2, 4), (x₂, y₂) = (6, a), (x₃, y₃) = (-1, 1), A = 9 in th formula

A=\frac{\left|x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right)\right|}{2}

9=\frac{\left|2\left(a-1\right)+6\left(1-4\right)+-1\left(4-a\right)\right|}{2}

Multiply both sides by 2

\frac{2\left|2\left(a-1\right)+6\left(1-4\right)-1\left(4-a\right)\right|}{2}=9\cdot \:2

simplify

\left|2\left(a-1\right)+6\left(1-4\right)-1\left(4-a\right)\right|=18

As the area is always positive.

so

2\left(a-1\right)+6\left(1-4\right)-1\cdot \left(4-a\right)=18

2\left(a-1\right)-18-\left(4-a\right)=18

Add 18 to both sides

2\left(a-1\right)-18-\left(4-a\right)+18=18+18

simplify

2\left(a-1\right)-\left(4-a\right)=36

3a-6=36

3a=42

Divide both sides by 3

\frac{3a}{3}=\frac{42}{3}

a=14

Thus, the value of a  = 14

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given that

let the height of the triangle be x

then the width is x-3

Area = \frac{height \times length}{2}

54 = x(x -3)

54 = x^2 -3x

x = 9

height = 9 cm

width = 6cm

Therefore, the height of a triangle is 9 cm.

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The value of f(19) in the given equation is 55.

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Given that:

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So;

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