The sides of a triangle are(x + 4)cm, xcm and (x - 4)cm, respectively If the cosine of the largest angle is \(\frac{1}{5}\), find the value of x

Ann_e

1 year ago

Given sides: x - 4, x, x + 4
Largest side = x + 4, so use cosine rule:

cos{largest angle} = {x^2 + (x - 4)^2 - (x + 4)^2}/{2x(x - 4)} = {1}/{5}

Simplify numerator:

x^2 + (x^2 - 8x + 16) - (x^2 + 8x + 16) = x^2 - 16x

So:

{x^2 - 16x}/{2x(x - 4)} = {1}/{5}=>5(x^2 - 16x) = 2x(x - 4)=>5x^2 - 80x = 2x^2 - 8x=>3x^2 - 72x = 0=>x(3x - 72) = 0=>x = 24

Answer: A. 24 cm

Mrnoah

1 year ago

Since the side facing angle B is the largest, (x + 4)cm, we can use the Law of Cosines to find the value of x.

cos B = (a² + c² - b²) / (2ac)

where:

a = x
c = x - 4
b = x + 4

Given cos B = 1/5 = 0.2:

0.2 = (x² + (x - 4)² - (x + 4)²) / (2x(x - 4))

Simplifying the equation:

0.2 = (x² + x² - 8x + 16 - x² - 8x - 16) / (2x² - 8x)

0.2 = (-16x) / (2x² - 8x)

Multiplying both sides by (2x² - 8x):

0.2(2x² - 8x) = -16x

Dividing both sides by 0.2:

2x² - 8x = -80x

Rearranging the equation:

2x² - 8x + 80x = 0

2x² + 72x = 0

Factoring out 2x:

2x(x + 36) = 0

This gives two possible values for x:

x = 0 or x = -36

However, since x represents the side length of a triangle, it cannot be negative.

So, we solve for x using the equation:

5(x - 16) = 2x - 8

Expanding and simplifying:

5x - 80 = 2x - 8

3x = 72

x = 24

Therefore, the value of x is indeed 24.

Mrnoah

1 year ago

The correct step is:

0.2 = (x² + (x - 4)² - (x + 4)²) / (2x(x - 4))

Simplifying the equation:

0.2 = (x² + x² - 8x + 16 - x² - 8x - 16) / (2x² - 8x)

0.2 = (-16x) / (2x² - 8x)

Multiplying both sides by (2x² - 8x):

0.2(2x² - 8x) = -16x

Dividing both sides by 0.2:

2x² - 8x = -80x

Rearranging the equation:

2x² + 72x = 0 is incorrect

Instead:

Multiply both sides of the equation by 5:

5(2x² - 8x) = 5(-80x)

10x² - 40x = -80x + 16x - 16

Combine like terms:

10x² - 40x + 80x = 16

10x² + 40x = 16 is incorrect

Instead:

5(x - 16) = 2x - 8

Expanding the equation:

5x - 80 = 2x - 8

Adding 80 to both sides:

5x = 2x + 72

Subtracting 2x from both sides:

3x = 72

Dividing both sides by 3:

x = 24