Questions: Quadrilateral BCDE is a kite. What is AD? AD=

Transcript text: Quadrilateral $B C D E$ is a kite. What is $A D$ ? \[ A D= \]

Solution

Solution Steps

Step 1: Analyze the given information

We are given that quadrilateral $BCDE$ is a kite. This means its diagonals are perpendicular, so segments $BD$ and $CE$ are perpendicular. We are also given that $AE = 48$ and $DE = 80$. We want to find $AD$.

Step 2: Use the Pythagorean theorem

Since $BD$ and $CE$ are perpendicular, triangle $ADE$ is a right triangle with the right angle at $A$. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In $\triangle ADE$, $DE$ is the hypotenuse, so we have: $AD^2 + AE^2 = DE^2$ We know $AE = 48$ and $DE = 80$. We can plug these values into the equation: $AD^2 + 48^2 = 80^2$

Step 3: Solve for AD

$AD^2 + 2304 = 6400$ $AD^2 = 6400 - 2304$ $AD^2 = 4096$ $AD = \sqrt{4096}$ $AD = 64$