Sure! Let's break down the given problem step-by-step to find the length of the kite's other diagonal.
### Step 1: Understand the Problem
Charlene has created a kite by joining two isosceles triangles along their bases. The triangles have the following properties:
- The legs of the first triangle are 10 inches each.
- The legs of the second triangle are 17 inches each.
- The common base for both triangles is 16 inches long.
### Step 2: Identify Necessary Elements
We need to determine the length of the kite's other diagonal. To calculate this, we must find the height of each isosceles triangle relative to the base.
### Step 3: Calculate Heights of Triangles
Since the base of both triangles is 16 inches, the height can be calculated by dropping a perpendicular from the apex of each triangle to the midpoint of the base (which is 8 inches from either end of the base).
#### Height of the First Triangle (Legs = 10 inches, Base = 16 inches):
- Given legs: \(10 \) inches
- Half of the base: \( 16 / 2 = 8 \) inches
Using the Pythagorean theorem in the right triangle:
[tex]\[
( \text{Leg} )^2 = (\text{Height})^2 + (\text{Half Base})^2
\][/tex]
[tex]\[
10^2 = \text{Height}^2 + 8^2
\][/tex]
[tex]\[
100 = \text{Height}^2 + 64
\][/tex]
[tex]\[
\text{Height}^2 = 100 - 64
\][/tex]
[tex]\[
\text{Height}^2 = 36
\][/tex]
[tex]\[
\text{Height} = 6 \, \text{inches}
\][/tex]
#### Height of the Second Triangle (Legs = 17 inches, Base = 16 inches):
- Given legs: \(17 \) inches
- Half of the base: \( 16 / 2 = 8 \) inches
Using the Pythagorean theorem in the right triangle:
[tex]\[
( \text{Leg} )^2 = (\text{Height})^2 + (\text{Half Base})^2
\][/tex]
[tex]\[
17^2 = \text{Height}^2 + 8^2
\][/tex]
[tex]\[
289 = \text{Height}^2 + 64
\][/tex]
[tex]\[
\text{Height}^2 = 289 - 64
\][/tex]
[tex]\[
\text{Height}^2 = 225
\][/tex]
[tex]\[
\text{Height} = 15 \, \text{inches}
\][/tex]
### Step 4: Calculate Length of the Kite's Other Diagonal
The kite's other diagonal is the sum of the heights of the two triangles:
[tex]\[
\text{Height of the first triangle} + \text{Height of the second triangle}
\][/tex]
[tex]\[
6 \, \text{inches} + 15 \, \text{inches} = 21 \, \text{inches}
\][/tex]
### Final Answer
The length of the kite's other diagonal is \( 21 \) inches. Hence, the correct choice is:
[tex]\[
\boxed{21 \, \text{inches}}
\][/tex]
