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The diagram shows a quadrilateral.
The lengths of the sides of the quadrilateral are written as expressions.
a Work out the lengths of the sides of the quadrilateral when 
a=-5 and 
b=2.
b Use your answers to part a to work out the perimeter of the quadrilate when 
a=-5 and 
b=2.
c Write an expression for the perimeter of the quadrilateral. Show that the expression can be simplified to 
a+4a^(2)+19 b.

The diagram shows a quadrilateral.\newlineThe lengths of the sides of the quadrilateral are written as expressions.\newlinea Work out the lengths of the sides of the quadrilateral when a=5 a=-5 and b=2 b=2 .\newlineb Use your answers to part a to work out the perimeter of the quadrilate when a=5 a=-5 and b=2 b=2 .\newlinec Write an expression for the perimeter of the quadrilateral. Show that the expression can be simplified to a+4a2+19b a+4 a^{2}+19 b .

Full solution

Q. The diagram shows a quadrilateral.\newlineThe lengths of the sides of the quadrilateral are written as expressions.\newlinea Work out the lengths of the sides of the quadrilateral when a=5 a=-5 and b=2 b=2 .\newlineb Use your answers to part a to work out the perimeter of the quadrilate when a=5 a=-5 and b=2 b=2 .\newlinec Write an expression for the perimeter of the quadrilateral. Show that the expression can be simplified to a+4a2+19b a+4 a^{2}+19 b .
  1. Calculate Side Lengths: Calculate the side lengths of the quadrilateral using the given values a=5a = -5 and b=2b = 2.
    Let's assume the expressions for the sides are:
    Side 11 = a+3a + 3
    Side 22 = 2ab2a - b
    Side 33 = 4b+14b + 1
    Side 44 = a22b+7a^2 - 2b + 7

    Plugging in the values:
    Side 11 = 5+3=2-5 + 3 = -2
    Side 22 = 2(5)2=122(-5) - 2 = -12
    Side 33 = 4(2)+1=94(2) + 1 = 9
    Side 44 = (5)22(2)+7=254+7=28(-5)^2 - 2(2) + 7 = 25 - 4 + 7 = 28
  2. Calculate Perimeter: Calculate the perimeter of the quadrilateral using the side lengths from Step 11.\newlinePerimeter = Side 1+Side 2+Side 3+Side 4\text{Side 1} + \text{Side 2} + \text{Side 3} + \text{Side 4}\newlinePerimeter = 2+(12)+9+28=23-2 + (-12) + 9 + 28 = 23
  3. Write Perimeter Expression: Write an expression for the perimeter of the quadrilateral and simplify it.\newlineExpression for perimeter = (a+3)+(2ab)+(4b+1)+(a22b+7)(a + 3) + (2a - b) + (4b + 1) + (a^2 - 2b + 7)\newlineCombine like terms:\newline=a+2a+a2+3b+4b2b+1+7= a + 2a + a^2 + 3 - b + 4b - 2b + 1 + 7\newline=a2+3a+b+11= a^2 + 3a + b + 11\newlineSimplified expression: a2+3a+b+11a^2 + 3a + b + 11

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