Question

quadrilateral qrst is a kite. what is su?
diagram of kite qrst with diagonals intersecting at u, rq=77, rs=85, su=?

Explanation:

Step1: Recall properties of a kite

In a kite, one of the diagonals is bisected by the other. Also, the triangles formed by the diagonals are right triangles? Wait, actually, in a kite, the diagonals are perpendicular, and one diagonal is bisected. Also, the two pairs of adjacent sides are equal. Here, we have triangle \( RUS \) which is a right triangle? Wait, no, in a kite, the diagonal that connects the vertices between the unequal sides is bisected and the diagonals are perpendicular. Wait, actually, in a kite, one diagonal is the perpendicular bisector of the other. So, \( RQ = RT \)? No, wait, in kite \( QRST \), the sides \( RQ \) and \( RT \)? Wait, no, the kite has two pairs of adjacent sides equal. So \( RQ = RT \)? No, maybe \( RQ = RS \) and \( QT = ST \)? Wait, looking at the lengths, \( RS = 85 \), \( RQ = 77 \)? Wait, no, maybe the diagonals intersect at \( U \), and in a kite, one diagonal is bisected, and the triangles formed are right triangles. Wait, actually, in a kite, the diagonal that is the axis of symmetry bisects the other diagonal and is perpendicular to it. So, triangle \( RUS \) is a right triangle with hypotenuse \( RS = 85 \), one leg \( RU = 77 \), and we need to find \( SU \).

Step2: Apply Pythagorean theorem

In right triangle \( RUS \), by Pythagoras theorem, \( SU = \sqrt{RS^2 - RU^2} \).

Given \( RS = 85 \), \( RU = 77 \).

So, \( SU = \sqrt{85^2 - 77^2} \)

Calculate \( 85^2 = 7225 \), \( 77^2 = 5929 \)

Then \( 7225 - 5929 = 1296 \)

\( \sqrt{1296} = 36 \)

Answer:

\( 36 \)