- Identify the kite's structure, where the diagonals are perpendicular and the common base is bisected.
- Use the Pythagorean theorem to find the length of the first segment of the diagonal, \(AM = \sqrt{10^2 - 8^2} = 6\) inches.
- Use the Pythagorean theorem to find the length of the second segment of the diagonal, \(CM = \sqrt{17^2 - 8^2} = 15\) inches.
- Sum the two segments to find the total length of the diagonal: \(AC = 6 + 15 = \boxed{21}\) inches.
Explanation
- Analyze the geometry of the kite.
A kite is formed by joining two isosceles triangles along a common base. Let the kite vertices be \(A, B, C, D\). The common base is diagonal \(BD = 16\) inches. Let \(AC\) be the other diagonal, which is the perpendicular bisector of \(BD\). Let \(M\) be the intersection of the diagonals \(AC\) and \(BD\). Since \(M\) is the midpoint of \(BD\), we have \(BM = MD = \frac{BD}{2} = \frac{16}{2} = 8\) inches. The triangle \(\triangle ABD\) is isosceles with legs \(AB = AD = 10\) inches, and the triangle \(\triangle CBD\) is isosceles with legs \(CB = CD = 17\) inches. We aim to find the total length of the diagonal \(AC = AM + CM\).
- Calculate the length of the diagonal segment AM.
In the right-angled triangle \(\triangle ABM\), the sides are the hypotenuse \(AB = 10\) and one leg \(BM = 8\). According to the Pythagorean theorem:
\[AM^2 + BM^2 = AB^2\]
\[AM^2 + 8^2 = 10^2\]
\[AM^2 + 64 = 100\]
\[AM^2 = 36\]
\[AM = \sqrt{36} = 6 \text{ inches.}\]
- Calculate the length of the diagonal segment CM.
In the right-angled triangle \(\triangle CBM\), the sides are the hypotenuse \(CB = 17\) and one leg \(BM = 8\). According to the Pythagorean theorem:
\[CM^2 + BM^2 = CB^2\]
\[CM^2 + 8^2 = 17^2\]
\[CM^2 + 64 = 289\]
\[CM^2 = 225\]
\[CM = \sqrt{225} = 15 \text{ inches.}\]
- Determine the total length of the diagonal AC.
The total length of the diagonal \(AC\) is the sum of its segments \(AM\) and \(CM\):
\[AC = AM + CM\]
\[AC = 6 + 15 = 21 \text{ inches.}\]
The diagonal length of the kite is \(\boxed{21}\) inches.
Examples
Imagine you are designing a large diamond-shaped kite for a school competition. You have two triangular frame pieces with specific leg lengths that must meet at a central crossbar (the base of 16 inches). By using this geometric method, you can accurately determine the total length of the vertical frame piece (the diagonal) needed to connect the tips of the triangles, ensuring your kite is perfectly symmetrical and aerodynamically balanced for flight.