### Video Transcript

π΄π΅πΆπ· is a kite, where π΄π΅ equals nine inches, π΅πΆ equals seven inches, and π΅π equals six inches. Determine the area of the kite to the nearest tenth.

We know that the two diagonals in a kite meet at right angles and that the area of a kite is equal to π multiplied by π divided by two, where π and π are the two diagonals of the kite. In this question, weβll calculate the area by multiplying the length of π΄πΆ by the length of π΅π· and then dividing by two.

We are told that the length of π΄π΅ is nine inches, the length of π΅πΆ is seven inches, and the length of π΅π is six inches. The length of π΅π is equal to the length of ππ·. As both of these are equal to six inches, the length of π΅π· is 12 inches. We can calculate the length of π΄π, labeled π₯, and πΆπ, labeled π¦, using the Pythagorean theorem.

This states that in any right-angled triangle, π squared plus π squared is equal to π squared, where π is the length of the hypotenuse and π and π are the lengths of the shorter sides of the triangle. If we consider triangle ππ΄π΅, then π΄π΅ is the hypotenuse as it is the longest side opposite the right angle.

Substituting in the lengths gives us π₯ squared plus six squared is equal to nine squared. Six squared is equal to 36, and nine squared is 81. Subtracting 36 from both sides of this equation gives us π₯ squared is equal to 45. Square rooting both sides gives us π₯ is equal to root 45. This can be simplified to three root five. The length of π΄π is three root five inches.

Repeating this process to calculate the length of πΆπ, we have π¦ squared plus six squared is equal to seven squared. Seven squared is 49. So we have π¦ squared plus 36 is equal to 49. Subtracting 36 from both sides gives us π¦ squared is equal to 13. Finally, square rooting both sides of this equation gives us π¦ is equal to root 13. The length of πΆπ is root 13.

We will now clear some space to calculate the area. The length π΄πΆ is equal to three root five plus root 13. We add the length of π΄π to the length of πΆπ. This needs to be multiplied by the length of π΅π·, which is 12. Once weβve multiplied these two lengths, we need to halve the answer or divide by two. Both two and 12 can be divided by two, leaving us with three root five plus root 13 multiplied by six.

We could distribute the parentheses at this stage by multiplying six by three root five and six by root 13. This is also called expanding the brackets and gives us an answer of 18 root five plus six root 13.

As we need to give our answer to the nearest tenth, we need to type this into the calculator. This gives us 61.8825 and so on. The second eight is the deciding number. If the deciding number is five or greater, we round up. This means that the area of the kite to the nearest tenth is 61.9 square inches. Our units for area will always be square units.