Unit 7
11 marks · HardTwo rectangles, A and B, are similar. The length of rectangle A is cm and its width is cm. The length of rectangle B is 12 cm and its width is 8 cm.
Show that the ratio of lengths to widths for both rectangles is equal. Write a quadratic equation for .
Solve the quadratic equation you have formed in part (a). Give both solutions for .
Explain which value of is valid for the dimensions of rectangle A, and justify your answer.
One side of a third rectangle C is cm, and this rectangle is similar to rectangles A and B. Find the width of rectangle C.
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
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(a) , so equation: or , which rearranges to · must show correct setup of similarity ratio, quadratic equation in standard form
(b) and after solving the quadratic · must show both roots shown in working
(c) is valid since a length cannot be negative; would give negative side lengths, which are not possible. · must show clear justification for physical dimensions
(d) Width of rectangle C is 4 cm. cm · must show using same scale as rectangle A, correct calculation of corresponding width
Common mistakes: Not setting up ratio as length:width for both rectangles.; Arithmetic errors expanding or rearranging.; Interpreting the quadratic's negative root as valid.
Show mark scheme and worked solution
Marking points
M1 (a) · 1 mark · States or uses the correct similarity ratio: or equivalent.
M2 (a) · 1 mark · Sets up and manipulates the ratio to form a quadratic equation.
A1 (a) · 1 mark · Correct quadratic equation in standard form.
M3 (b) · 1 mark · Attempts to solve the quadratic (by factorisation, formula, or completing the square).
A2 (b) · 1 mark · One correct root found.
A3 (b) · 1 mark · Both roots found with correct working.
M4 (c) · 1 mark · Deduces which root is valid (based on context/sense). (follow-through)
A4 (c) · 1 mark · Gives clear reason or justification for only accepting . (follow-through)
M5 (d) · 1 mark · Uses correct scale relationship to work out width for rectangle C, using . (follow-through)
A5 (d) · 2 marks · Correct calculation of width for rectangle C ( cm). Award both marks for full method and correct value; award one for a correct method with wrong answer due to a slip. (follow-through)
Worked solution
a) In similar rectangles, the ratio of length to width must be the same:
Cross-multiply:
Bring all terms to one side:
So , but also note that the quadratic form (if expanded differently) can be:
Alternatively, if starting from , cross-multiplying gives , etc.
b) The equation above simplifies to . If written in standard quadratic form, the equation to solve would be , whose solutions by the quadratic formula are:
(In context, only gives positive rectangle sides.)
c) is valid because rectangles cannot have negative or zero length, but makes the width 0 and (from other possible rearrangements) is not physically meaningful. So accept only .
d) Rectangle C has length cm. To find its width (let it be ), use the similarity ratio:
So cm. But since both A and C are similar and have lengths 6 and 12, their widths are 4 and 8 (since A: width is ). If we try length of C, , so the width for the same increase should be cm, which matches the ratio.
If you interpret 'rectangle C has side x+6' as width (not length), apply the same method.