We are told that the measure of AC is 7. On the diagram we can see that segments AC and CE are congruent, which means that their measures are the same. Hence, CE is also 7.
CE=7
It is also given that DC measures 9. From the diagram we know that DC and CB are congruent segments. This allows us to conclude that CB is also 9.
CB=9
Let's add this information to the diagram.
Also, we can see that ∠ BCA and ∠ DCE are vertical angles. By the Vertical Angles Theorem, vertical angles are always congruent. We got that △ ABC and △ CDE have two pairs of congruent sides which form two congruent angles ∠ BCA and ∠ DCE.
ccc
△ ABC & & △ CDE [0.2em]
AC &≅ & CE
BC & ≅ & CD
∠ BCA & ≅ & ∠ DCE
Therefore, by the Side-Angle-Side Theorem
triangles △ ABC and △ CDE are congruent.
△ ABC ≅ △ CDE
Sides AB and DE are also congruent segments and have the same measure. Let this measure be x. To find the value of x, let's recall what the Triangle Inequality Theorem states.
The sum of the lengths of any two
sides of a triangle must be greater
than the length of the third side.
Using this theorem, we can form three inequalities for the triangle △ ABC, which are also true for △ CDE.
AB+BC>AC
AB+AC>BC
BC+AC>AB
Let's substitute the measures of the sides and solve these inequalities for x.
Inequality
Substitute
Solution Set
AB+BC>AC
x+9>7
x>- 2
AB+AC>BC
x+7>9
x>2
BC+AC>AB
9+7>x
x<16
Now we can graph these solutions sets and find the common solutions.
We got that the value of x, which is the measure of AB and DE, must be greater than 2 and less than 16. Let's use these values to find the smallest and greatest value of the perimeter of ABCDE. We will find the perimeter of △ ABC and then multiply by 2, because ABCDE consists of two triangles of the same perimeter.
Smallest: P_(ABCDE)>2(2+9+7)=36
Greatest: P_(ABCDE)<2(16+9+7)=64
Therefore, the range for the possible perimeters of ABCDE is from 36 to 64.
