𝐴𝐵𝐶𝐷 is a kite, where 𝐴𝐵 equals nine inches, 𝐵𝐶 equals seven inches, and 𝐵𝑂 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 𝑝 multiplied by 𝑞 divided by two, where 𝑝 and 𝑞 are the two diagonals of the kite. In this question, we’ll calculate the area by multiplying the length of 𝐴𝐶 by the length of 𝐵𝐷 and then dividing by two.
We are told that the length of 𝐴𝐵 is nine inches, the length of 𝐵𝐶 is seven inches, and the length of 𝐵𝑂 is six inches. The length of 𝐵𝑂 is equal to the length of 𝑂𝐷. As both of these are equal to six inches, the length of 𝐵𝐷 is 12 inches. We can calculate the length of 𝐴𝑂, labeled 𝑥, and 𝐶𝑂, labeled 𝑦, using the Pythagorean theorem.
This states that in any right-angled triangle, 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the length of the hypotenuse and 𝑎 and 𝑏 are the lengths of the shorter sides of the triangle. If we consider triangle 𝑂𝐴𝐵, then 𝐴𝐵 is the hypotenuse as it is the longest side opposite the right angle.
Substituting in the lengths gives us 𝑥 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 𝑥 squared is equal to 45. Square rooting both sides gives us 𝑥 is equal to root 45. This can be simplified to three root five. The length of 𝐴𝑂 is three root five inches.
Repeating this process to calculate the length of 𝐶𝑂, we have 𝑦 squared plus six squared is equal to seven squared. Seven squared is 49. So we have 𝑦 squared plus 36 is equal to 49. Subtracting 36 from both sides gives us 𝑦 squared is equal to 13. Finally, square rooting both sides of this equation gives us 𝑦 is equal to root 13. The length of 𝐶𝑂 is root 13.
We will now clear some space to calculate the area. The length 𝐴𝐶 is equal to three root five plus root 13. We add the length of 𝐴𝑂 to the length of 𝐶𝑂. This needs to be multiplied by the length of 𝐵𝐷, which is 12. Once we’ve 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.