Problem
Name the unknown with a symbol and build an equation
A rhombus has a known area and one known diagonal. We need the difference between its two diagonals.
ExpressionsGeometry
Your answer
How to solve
Strategy Convert to Algebra — The area rule already ties the two diagonals together, so name the missing one and the rule becomes an equation with a single unknown. Solve it, then remember the question asks for the difference, not the diagonal.
1STEP 1
Write the area rule with the unknown in it
A rhombus's area is its two diagonals multiplied and halved. Call the missing diagonal ■ and write 1 2/5 as 7/5.
7/5 × ■ ÷ 2 = 14/25
One equation, one unknown.
6.G.A.1Convert To Algebra2STEP 2
Undo the halving
Double both sides to get rid of the division by 2.
14/25 × 2 = 28/25
The two diagonals multiply to 28/25.
6.EE.B.7Identify SubproblemsDoubling both sides removes the halving from the area rule.
Why?
A rhombus is half of the rectangle its two diagonals span, so its area is that product halved.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Multiplying by two undoes dividing by two, so doing it to both sides leaves the equality standing.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
3STEP 3
Divide out the known diagonal
Dividing by 7/5 is multiplying by 5/7.
28/25 × 5/7 = 4/5
The other diagonal is 4/5 cm.
6.NS.A.1Work Backwards4STEP 4
Take the difference
Both diagonals are now in fifths, so subtract straight across.
7/5 - 4/5 = 3/5
They differ by 3/5 cm.
6.EE.B.7Identify SubproblemsAnswer
cm
Put both diagonals back in: 7/5 × 4/5 ÷ 2 = 28/50, which reduces to 14/25 cm².
Takeaway
An area rule with one unknown in it is already an equation -- name the unknown and solve.
- Write the area rule with the unknown in it
- Undo the halving
- Divide out the known diagonal
- Take the difference