Hi there! :)
<u>Answer:</u>
Length of the second diagonal: 54 cm
Area of the rhombus: 1,944 cm²
Step-by-step explanation:
1) To find the value of the second diagonal, you first need to know some important properties of rhombuses:
Since the perimeter is 180, and by definition a rhombus has four congruent sides (of equal length), each side length of the rhombus is equal to 45.
The diagonals of rhombuses also form four right triangles, with hypotenuses equal to the side length of the rhombus and legs equal to half the lengths of the diagonals. In this case one hypotenuse would be equal to 45 and one leg would be equal to 36 (72 ÷ 2 = 36).
We can therefore use the Pythagorean Theorem to solve for one-half of the unknown diagonal:
c² = a² + b² → Where "c" is the hypotenus & "a" and "b" are the two other sides.
<u>45² = 36² + b²</u> → Where "b" is the half of the unknown diagonal.
We can now solve this equation for "b":
<u>45²</u> = <u>36²</u> + b²
45² = <u>2,025</u>
36² = <u>1,296</u>
2,025 = <u>1,296</u> + b²
Subtract "1,296" from each side of the equation → 2,025 - 1,296 = <u>729</u>
729 = b<u>²</u>
Square root on each side of the equation → √729 = <u>27</u>
<u>27 = b</u>
Since "b" represents half of the unknown diagonal, we need to multiply by 2 to find the full length of the diagonal:
27 × 2 = <u>54 ⇒ Length of the other diagonal.</u>
2) To find the area of the rhombus, you need to multiply the length of the 2 diagonals and divide the answer by 2:
<u>(72 × 54)</u> ÷ 2 = A
Multiply "72" & "54" together → 72 × 54 = <u>3,888</u>
<u>3,888 ÷ 2</u> = A
<u>1,944 = A</u>
There you go! I really hope this helped, if there's anything just let me know! :)
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