Find the area in cm\(^2\) of a rhombus whose side length is 29 cm and whose diagonals differ in length by 2 cm.
Answer (2)
( 2 d ) 2 + ( 2 d + 2 ) 2 = 2 9 2 4 d 2 + 4 d 2 + 4 d + 4 = 841 / ⋅ 4 d 2 + d 2 + 4 d + 4 = 3364 2 d 2 + 4 d + 4 − 3364 = 0 2 d 2 + 4 d − 3360 = 0 / : 2 d 2 + 2 d − 1680 = 0
a = 1 ; b = 2 ; c = − 1680 Δ = b 2 − 4 a c → Δ = 2 2 − 4 ⋅ 1 ⋅ ( − 1680 ) = 4 + 6720 = 6724 Δ = 6724 = 82 d 1 = 2 a − b − Δ → d 1 = 2 ⋅ 1 − 2 − 82 < 0 d 2 = 2 a − b + Δ → d 2 = 2 ⋅ 1 − 2 + 82 = 2 80 = 40 ( c m ) d = 40 c m ; d + 2 = 42 c m A r = 2 d ( d + 2 ) → A r = 2 40 ⋅ 42 = 840 ( c m 2 )
The area of the rhombus is calculated to be 840 cm² by using the properties of its sides and diagonals. We set up a quadratic equation based on the given conditions, solved for the lengths of the diagonals, and then applied the area formula. The final result confirms the area as 840 cm².
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