A kite-shaped quadrilateral has sides of lengths $5$ cm and $7$ cm, and the shorter diagonal is $8$ cm. What is the length of the kite's longer diagonal?

• A. $12$ cm
• B. $14$ cm
• C. $9$ cm
• D. $10$ cm

Answer: (A)

Verified step by step guidance

1. Recall that a kite is a quadrilateral with two pairs of adjacent sides equal. Label the kite's sides as 5 cm and 7 cm, with the shorter diagonal given as 8 cm.
2. Understand that the diagonals of a kite are perpendicular to each other. Let the longer diagonal be denoted as \(d\). The diagonals intersect at right angles, dividing the kite into four right triangles.
3. Split the kite along the diagonals and use the Pythagorean theorem on the right triangles formed. Let the half-lengths of the diagonals be \(\frac{8}{2} = 4\) cm and \(\frac{d}{2}\) respectively.
4. Set up equations for the two right triangles using the side lengths: for the triangle with sides 5 cm and half diagonals, \$5^2 = 4^2 + \left(\frac{d}{2}\right)^2\(, and for the triangle with sides 7 cm and half diagonals, \)7^2 = 4^2 + \left(\frac{d}{2}\right)^2$.
5. Solve these equations to find the value of \(d\), the longer diagonal, by isolating \(\frac{d}{2}\) and then doubling it to get the full length.