Identify the whole-number measure of segment AD¯¯¯¯¯¯¯¯ that would create △ABD and △ACD as shown.

Question

An illustration shows a rhombus where the vertices are labeled as follows: A on the top vertex, B on the left, C on the right, and D at the bottom. A dashed line is drawn from A to D. The sides are labeled as follows: A C is labeled as 5, A B as 4, B D as 8, and C D as 15.

(1 point)
Responses

12
12 - no response given

9
9 - incorrect

5
5 - no response given

11

Answer 1

To solve this problem, we need to analyze the properties of the given rhombus and the information provided about the segments.

1. Since we know that a rhombus has all its sides equal, we would determine that the lengths of all sides of rhombus ABCD should be the same.
2. The sides are labeled as follows: \( AC = 5 \), \( AB = 4 \), \( BD = 8 \), and \( CD = 15 \).

From the given segment lengths, the sides \( AC \) and \( AB \) do not match the lengths of sides \( BD \) and \( CD \). Hence, it suggests that we might not be working with a perfect rhombus. Instead, the triangle lengths may provide more information about what \( AD \) should be for the triangles \( \triangle ABD \) and \( \triangle ACD \).

Now let's use the triangle inequalities on each triangle to determine constraints on \( AD \):

1. For triangle \( ABD \):
   • \( AB + BD > AD \)
   • \( 4 + 8 > AD \) which implies \( AD < 12 \)
   • \( AB + AD > BD \)
   • \( 4 + AD > 8 \) which implies \( AD > 4 \)
   • \( BD + AD > AB \)
   • \( 8 + AD > 4 \) which is always true given the other conditions.

So from triangle \( ABD \), we get:

• \( 4 < AD < 12 \)

2. For triangle \( ACD \):
   • \( AC + CD > AD \)
   • \( 5 + 15 > AD \) which implies \( AD < 20 \) (which doesn't add new information given the previous constraint)
   • \( AC + AD > CD \)
   • \( 5 + AD > 15 \) which implies \( AD > 10 \)
   • \( CD + AD > AC \)
   • \( 15 + AD > 5 \) which is always true given the other conditions.

So from triangle \( ACD \), we get:

• \( AD > 10 \)

Combining the results from both triangles, we find:

• From triangle \( ABD \): \( 4 < AD < 12 \)
• From triangle \( ACD \): \( AD > 10 \)

Thus, combining these inequalities, we have:

• \( 10 < AD < 12 \)

As \( AD \) must be a whole number, the only possible value for \( AD \) is 11.

Therefore, the measure of segment \( AD \) is 11.