Depth · one concept from first principles · step 2 of 4
The rhombus at the right has an area of . If one of its diagonals is long, what is the difference between the lengths of the two diagonals, in ?
A rhombus's area is its two diagonals multiplied and halved. Call the missing diagonal ■ and write 1 2/5 as 7/5.
One equation, one unknown.
6.G.A.1Convert To AlgebraDouble both sides to get rid of the division by 2.
The two diagonals multiply to 28/25.
6.EE.B.7Identify SubproblemsDoubling both sides removes the halving from the area rule.
A rhombus is half of the rectangle its two diagonals span, so its area is that product halved.
Multiplying by two undoes dividing by two, so doing it to both sides leaves the equality standing.
Dividing by 7/5 is multiplying by 5/7.
The other diagonal is 4/5 cm.
6.NS.A.1Work BackwardsBoth diagonals are now in fifths, so subtract straight across.
They differ by 3/5 cm.
6.EE.B.7Identify SubproblemsAn area rule with one unknown in it is already an equation -- name the unknown and solve.
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The same shape shows up later on the AMC, as Arithmetic Expression Decomposition, Branching State Enumeration, Multiplicative Ratio with Integrality Constraints. Why the shapes repeat →