Question

find the perimeter of the following kite, if qs = 10, pl = 12, and lr = 9. round all answers to the nearest tenth.

Explanation:

Step1: Analyze kite properties

In a kite, the diagonals are perpendicular, and one diagonal is bisected by the other. So, \( QL = LS=\frac{QS}{2}=\frac{10}{2} = 5 \), \( PL = LR = 12? \) Wait, no, the problem says \( PL = 12 \) and \( LR = 9 \)? Wait, no, maybe I misread. Wait, the kite has diagonals \( PR \) and \( QS \), intersecting at \( L \). In a kite, two pairs of adjacent sides are equal. Also, diagonals intersect at right angles. So \( PL + LR = PR \), so \( PR=12 + 9=21 \)? Wait, no, the problem says \( PL = 12 \), \( LR = 9 \), so \( PR=PL + LR=21 \), and \( QS = 10 \), so \( QL = LS = 5 \) (since diagonals in a kite: one diagonal is bisected, the other is not necessarily, but the perpendicular one? Wait, actually, in a kite, one diagonal is the perpendicular bisector of the other. So if \( QS \) and \( PR \) are diagonals, intersecting at \( L \), then \( QL = LS \) (since \( QS \) is bisected by \( PR \) at right angles), and \( PL \) and \( LR \) are segments of \( PR \). So \( QL = 5 \), \( PL = 12 \), \( LR = 9 \).

Step2: Find the lengths of the sides

The sides of the kite: \( PQ = PS \) and \( RQ = RS \) (adjacent sides equal). To find \( PQ \), use the right triangle \( PLQ \): \( PQ=\sqrt{PL^{2}+QL^{2}}=\sqrt{12^{2}+5^{2}}=\sqrt{144 + 25}=\sqrt{169}=13 \). To find \( RQ \), use right triangle \( RLQ \): \( RQ=\sqrt{LR^{2}+QL^{2}}=\sqrt{9^{2}+5^{2}}=\sqrt{81 + 25}=\sqrt{106}\approx10.3 \).

Step3: Calculate the perimeter

Perimeter of kite is \( 2\times(PQ + RQ)=2\times(13 + 10.3)=2\times23.3 = 46.6 \)? Wait, no, wait: \( PQ = PS = 13 \), \( RQ = RS\approx10.3 \). So perimeter is \( PQ + PS + RQ + RS = 13 + 13 + 10.3 + 10.3 = 26 + 20.6 = 46.6 \). Wait, but let's recalculate \( RQ \): \( \sqrt{9^{2}+5^{2}}=\sqrt{81 + 25}=\sqrt{106}\approx10.2956\approx10.3 \). Then \( PQ=\sqrt{12^{2}+5^{2}}=\sqrt{144 + 25}=\sqrt{169}=13 \). So two sides are 13, two sides are \( \approx10.3 \). So perimeter is \( 2\times13 + 2\times10.3 = 26 + 20.6 = 46.6 \).

Wait, but maybe I made a mistake in the diagonals. Wait, the problem says \( PL = 12 \), \( LR = 9 \), so \( PR = 21 \), and \( QS = 10 \), so \( QL = 5 \), \( LS = 5 \). Then triangles \( PLQ \) (right triangle, legs 12 and 5, hypotenuse \( PQ \)), \( RLQ \) (legs 9 and 5, hypotenuse \( RQ \)), \( PLS \) (same as \( PLQ \), so \( PS = PQ = 13 \)), \( RLS \) (same as \( RLQ \), so \( RS = RQ\approx10.3 \)). So perimeter is \( 13 + 13 + 10.3 + 10.3 = 46.6 \).

Answer:

The perimeter of the kite is \(\boxed{46.6}\)