To determine the length of the kite's shorter diagonal, we need to focus on the heights of the two isosceles triangles that share a common base.
1. Examine the triangles:
- The base of each triangle is 80 inches.
- The legs of the upper triangle measure 41 inches each.
- The legs of the lower triangle measure 50 inches each.
2. Determine the height of the upper triangle:
In an isosceles triangle, if we drop a perpendicular from the vertex opposite the base to the midpoint of the base, it will split the base into two equal segments of 40 inches each. We can use the Pythagorean theorem to find the height (h).
For the upper triangle:
[tex]\[
(41)^2 = (40)^2 + h_{upper}^2
\][/tex]
[tex]\[
1681 = 1600 + h_{upper}^2
\][/tex]
[tex]\[
h_{upper}^2 = 1681 - 1600
\][/tex]
[tex]\[
h_{upper}^2 = 81
\][/tex]
[tex]\[
h_{upper} = \sqrt{81} = 9 \text{ inches}
\][/tex]
3. Determine the height of the lower triangle:
Similarly, for the lower triangle:
[tex]\[
(50)^2 = (40)^2 + h_{lower}^2
\][/tex]
[tex]\[
2500 = 1600 + h_{lower}^2
\][/tex]
[tex]\[
h_{lower}^2 = 2500 - 1600
\][/tex]
[tex]\[
h_{lower}^2 = 900
\][/tex]
[tex]\[
h_{lower} = \sqrt{900} = 30 \text{ inches}
\][/tex]
4. Determine the kite's shorter diagonal:
The shorter diagonal of the kite corresponds to the height of the upper triangle, which we found to be 9 inches.
Therefore, the length of the kite's shorter diagonal is [tex]\( 9 \)[/tex] inches. This matches none of the given choices directly, as shown below:
- 30 inches (Incorrect)
- 39 inches (Incorrect)
- [tex]\( 10 \sqrt{39} \)[/tex] inches (Incorrect)
- [tex]\( 10 \sqrt{11} \)[/tex] inches (Incorrect)
The actual length of the kite's shorter diagonal is 9 inches.
