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?
It should be easy to see that
AB = AC = AD
let each one be x
in triangle ADC
2x + 7 = 51
2x= 44
x = 22
in triangle ABC
2x + BC = 59
44 + BC = 59
BC = 15
To find the value of BC, we can use the information given about the perimeters of triangles ADC and ABC.
Let's break down the problem step by step:
1. We are given that CD = 7. Let's label this information on the diagram:
D
|\
| \
| \
7 | \
| \
|_____\
C B
2. We are given that ∠ADC = ∠ACD and ∠ACB = ∠ABC. This means that triangle ADC is isosceles with AD = AC, and triangle ABC is isosceles with AB = AC.
3. We are given that the perimeter of triangle ADC is 51. Since it is isosceles, this implies that AD + AC + CD = 51. But since AD = AC and CD = 7, we can replace AD and CD with AC and 7, respectively. Therefore, we have:
AC + AC + 7 = 51
2AC + 7 = 51
2AC = 51 - 7
2AC = 44
AC = 44/2
AC = 22
Therefore, AC = 22.
4. Similarly, since the perimeter of triangle ABC is 59 (which is larger than 51), we can apply the same logic. Since AB = AC, the perimeter equation becomes:
AB + AC + BC = 59.
Since we found that AC = 22, we can substitute this into the equation:
AB + 22 + BC = 59.
Rearranging the equation:
AB + BC = 59 - 22
AB + BC = 37.
But from the given information, we know that ∠ACB = ∠ABC, which means AB = BC. Therefore, we can substitute AB with BC:
BC + BC = 37
2BC = 37
BC = 37/2
BC = 18.5.
Therefore, BC = 18.5.
In summary, the value of BC is 18.5.