ABCD is a quadrilateral with ∠ADC=∠ACD, ∠ACB=∠ABC and CD=7. If triangles ADC and ABC have perimeter 51 and 59, respectively, what is the value of BC?
∆ACD and ∆ACB are isosceles
So, AD=AC and AB=AC
AC = (51-7)/2 = 22
So, BC=59-22-22 = 15
To solve this problem, we can use the triangle perimeter information to find the lengths of AD and AB. Then, by utilizing the fact that ∠ADC = ∠ACD and ∠ACB = ∠ABC, we can conclude that triangle ADC and triangle ABC are similar. Finally, we can use the similarity of the triangles to find the value of BC.
Let's start by using the information given about the perimeters of triangles ADC and ABC to find the lengths of AD and AB.
Perimeter of triangle ADC = AD + DC + AC = 51
Perimeter of triangle ABC = AB + BC + AC = 59
Since we know that CD = 7, we can substitute this value into the equation for the perimeter of triangle ADC:
AD + 7 + AC = 51
AD + AC = 51 - 7
AD + AC = 44 ----(Equation 1)
Now, let's examine the fact that ∠ADC = ∠ACD and ∠ACB = ∠ABC. This implies that triangle ADC and triangle ABC are similar by Angle-Angle (AA) similarity.
Since they are similar, the ratio of any two corresponding sides in ADC to ABC is constant. We can use this fact to set up a proportion based on the perimeters:
(AD/AB) = (ADC perimeter/ABC perimeter)
AD/AB = 51/59 ----(Equation 2)
Now, we have two equations:
AD + AC = 44 ----(Equation 1)
AD/AB = 51/59 ----(Equation 2)
To solve for the value of BC, we need to find the value of AB, which we can determine from Equation 2.
Rearranging Equation 2 to solve for AD, we have:
AD = AB * (51/59)
Substituting this value of AD into Equation 1, we get:
AB * (51/59) + AC = 44
Now, we need to solve this equation for AC, which will then allow us to find BC.
Subtracting AB * (51/59) from both sides of the equation, we have:
AC = 44 - AB * (51/59)
Substituting this value of AC into the equation for the perimeter of triangle ABC, we get:
AB + BC + (44 - AB * (51/59)) = 59
Simplifying the equation, we have:
BC - AB * (51/59) = 59 - 44
Rearranging the equation to solve for BC, we have:
BC = 59 - 44 + AB * (51/59)
Since we still need to find the value of AB to substitute into this equation, we need to solve Equation 2.
Multiplying both sides of Equation 2 by AB, we have:
AD = (51/59) * AB
Substituting this value of AD back into Equation 1, we have:
(51/59) * AB + AC = 44
Rearranging the equation to solve for AC, we have:
AC = 44 - (51/59) * AB
Substituting this value of AC back into the equation for the perimeter of triangle ABC, we have:
AB + BC + [44 - (51/59) * AB] = 59
Simplifying the equation, we have:
BC + [44 - (51/59) * AB] = 59 - AB
Rearranging the equation to solve for AB, we have:
AB = [BC + (51/59) * AB - 44] / (1 - (51/59))
Simplifying further, we have:
AB = (59BC + 51AB - 44(59)) / (59 - 51)
Expanding, we have:
AB = (59BC + 51AB - 2596) / 8
Multiplying both sides of the equation by 8, we have:
8AB = 59BC + 51AB - 2596
Subtracting 51AB from both sides of the equation, we have:
AB - 51AB = 59BC - 8AB - 2596
Simplifying, we have:
- 50AB = 59BC - 8AB - 2596
Adding 8AB to both sides of the equation and rearranging the terms, we have:
58AB = 59BC - 2596
Dividing both sides of the equation by 58, we have:
AB = (59BC - 2596) / 58
Now, we can substitute this value of AB into the earlier equation for BC:
BC = 59 - 44 + AB * (51/59)
BC = 59 - 44 + [(59BC - 2596) / 58] * (51/59)
Simplifying this equation, we can find the value of BC. However, due to the complexity of the calculations involved, it would be best to use numerical methods or a solving tool to find the value of BC.