Question

quadrilateral rstu is a kite. what is qs?
qs =
submit

Explanation:

Step1: Recall kite diagonal property

One diagonal of a kite is bisected by the other, so the segment $QT = 39$ is equal to the segment $QR$. Thus, $QR = 39$.

Step2: Calculate length of QS

QS is the sum of $QT$ and $TS$? No, correct: QS is $QR + QS$? No, correct: QS is $QT + QS$? Wait, no: QS is composed of $QT$ and $QS$? No, the full diagonal QS is $QT + SQ$? No, looking at the diagram: Q is the intersection point. The segment $TS=89$ is a side, wait no: in kite, the diagonals are perpendicular, and the sides adjacent to each other are equal. Wait, no, the right triangle QTS: $QT=39$, $TS=89$, so we find $QS$? No, wait, Q is the intersection, so triangle QTS is right-angled at Q. So we use Pythagoras to find $QS$? No, wait, $TS$ is the side of the kite, $QT$ is part of the diagonal. So:
Wait, correct step: In right $\triangle QTS$, right-angled at Q, find $QS$.

Step1: Identify right triangle sides

In $\triangle QTS$, $\angle TQS = 90^\circ$, $QT=39$, $TS=89$.

Step2: Apply Pythagorean theorem

$$QS = \sqrt{TS^2 - QT^2}$$
$$QS = \sqrt{89^2 - 39^2}$$
$$QS = \sqrt{(89-39)(89+39)}$$
$$QS = \sqrt{50 \times 128}$$
$$QS = \sqrt{6400}$$
$$QS = 80$$

Answer:

$80$