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?
Angle adc=110
To find the value of BC, we can use the fact that the perimeters of triangles ADC and ABC are given as 51 and 59, respectively.
Let's break down the solution step by step:
Step 1: Given information:
- Quadrilateral ABCD with ∠ADC = ∠ACD and ∠ACB = ∠ABC.
- CD = 7.
- Perimeter of triangle ADC is 51.
- Perimeter of triangle ABC is 59.
Step 2: Perimeter of triangle ADC:
The perimeter of a triangle is the sum of the lengths of its three sides. Since ADC is a triangle, we can write the equation:
AD + DC + AC = 51
Step 3: Perimeter of triangle ABC:
Similarly, we can write the equation for the perimeter of triangle ABC:
AB + BC + AC = 59
Step 4: Applying the given information:
Since ABCD is a quadrilateral and ∠ADC = ∠ACD, we can deduce that AD and AC are equal in length. Therefore, we can substitute AC for both AD and AC in the equations from steps 2 and 3.
Step 5: Substitution:
Using the substitution, our equations become as follows:
AC + DC + AC = 51
AB + BC + AC = 59
Step 6: Simplification:
Simplifying the equations, we get:
2AC + DC = 51
AB + BC + AC = 59
Step 7: Solve for AC:
From the first equation, rearrange it to solve for AC:
2AC = 51 - DC
AC = (51 - DC)/2
Step 8: Substitute in the second equation:
Substituting the value of AC into the second equation:
AB + BC + (51 - DC)/2 = 59
Step 9: Find BC:
Simplifying the equation, we get:
AB + BC + 51/2 - DC/2 = 59
AB + BC = 59 - 51/2 + DC/2
AB + BC = 59 - 51/2 + 7/2
AB + BC = 59 - 44/2
AB + BC = 59 - 22
AB + BC = 37
Therefore, the value of BC is 37.