Exercise 25 Page 457

We are told that the polygons are similar, and we want to find the value of x for the missing side lengths.

Since the figures are similar, their corresponding sides are also similar. Let's identify corresponding sides.

• There is only one side on each polygon included between two acute angles. These are corresponding sides.
• There is only one shorter side on each polygon included between the right angle and acute angle. Thus, they are also corresponding sides.
• Similarly, there is only one longer side on each polygon included between the right angle and acute angle. Thus, they are also corresponding sides.
  
  The ratio between corresponding sides is the same for all pairs of corresponding sides. 2x - 4/24=39/x + 6 Let's solve the above equation using Cross Product Property.
  2x - 4/24=39/x + 6

  ▼

  Solve for x

  CrossMult

  Cross multiply

  (2x-4)(x+6)=24* 39

  Multiply

  Multiply

  (2x-4)(x+6)=936

  Distr

  Distribute (x+6)

  2x(x+6)-4(x+6)=936

  Distr

  Distribute 2x & - 4

  2x^2+12x-4x-24=936

  SubTerm

  Subtract term

  2x^2+8x-24=936

  SubEqn

  LHS-936=RHS-936

  2x^2+8x-960=0

  FactorOut

  Factor out 2

  2(x^2+4x-480)=0

  DivEqn

  .LHS /2.=.RHS /2.

  x^2 + 4x - 480 = 0
  Notice that we have to solve a quadratic equation. We will do it by using the Quadratic Formula. ax^2+bx+c=0 ⇔ x=- b±sqrt(b^2-4ac)/2a To use this formula, we first need to identify the values of a, b, and c in our equation. x^2 + 4x - 480 = 0 ⇔ 1x^2+ 4x+( - 480)=0 We can see above that a= 1, b= 4, and c= - 480. Now we can substitute these values into the Quadratic Formula.
  x=- b±sqrt(b^2-4ac)/2a

  SubstituteValues

  Substitute values

  x=- 4±sqrt(4^2-4( 1)( - 480))/2( 1)

  ▼

  Simplify

  CalcPow

  Calculate power

  x=- 4±sqrt(16-4(1)(- 480))/2(1)

  MultByOne

  a * 1=a

  x=- 4±sqrt(16-4(- 480))/2

  MultNegNegOnePar

  - a(- b)=a* b

  x=- 4±sqrt(16+1920)/2

  AddTerms

  Add terms

  x=- 4±sqrt(1936)/2

  CalcRoot

  Calculate root

  x=- 4 ± 44/2
  Remember that x must be positive number since expressions including it, 2x-4 and x+6, must be positive. x = - 4 + 44/2 ⇔ x = 20