Answer:
kite; P=45–√+413−−√ units; A=32 units2
Explanation:
Find the length of each side using the Distance Formula.
Find the length of side HA.
HA=(−10−(−6))2+(4−(−2))2−−−−−−−−−−−−−−−−−−−−−−−√
=(−4)2+(6)2−−−−−−−−−−√
=16+36−−−−−−√
=52−−√=213−−√
Find the length of side AT.
AT=(−6−(−10))2+(6−4)2−−−−−−−−−−−−−−−−−−−−√
=(4)2+(2)2−−−−−−−−−√
=16+4−−−−−√
=20−−√=25–√
Find the length of side TC.
TC=(−2−(−6))2+(4−6)2−−−−−−−−−−−−−−−−−−−√
=(4)2+(−2)2−−−−−−−−−−√
=16+4−−−−−√
=20−−√=25–√
Find the length of side HC.
HC=(−2−(−6))2+(4−(−2))2−−−−−−−−−−−−−−−−−−−−−−√
=(4)2+(6)2−−−−−−−−−√
=16+36−−−−−−√
=52−−√=213−−√
Therefore, HA=HC=213−−√ and AT=TC=25–√.
Since there are two pairs of congruent consecutive sides, HATC is a kite by definition.
Find the perimeter of the kite as the sum of the four sides.
P=213−−√+25–√+25–√+213−−√=45–√+413−−√ units
To find the area of the kite, first find the lengths of the diagonals AC and TH using the Diatance Formula.
The figure shows the same quadrilateral H A T C as in the beginning of the task. Diagonal A C is perpendicular to diagonal T H.
Find the length of diagonal AC.
AC=(−2−(−10))2+(4−4)2−−−−−−−−−−−−−−−−−−−−√
=(8)2+(0)2−−−−−−−−−√
=64−−√=8
Find the legth of diagonal TH.
TH=(−6−(−6))2+(−2−6)2−−−−−−−−−−−−−−−−−−−−−√
=(0)2+(−8)2−−−−−−−−−−√
=64−−√=8
Find the area of the kite using the formula for the area of a kite, A=12d1d2.
Substitute 8 for d1 and 8 for d2.
A=12(8)(8)=32 units2
Therefore, the perimeter of the kite is 45–√+413−−√ units and the area is 32 units2.