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TMUA 2023 · Paper 1 · Question 5 of 20

TMUA 2023 Paper 1 Question 5

Coordinate geometry — Inscribed rectangle · quadratic in a length. Try it first; the answer and a full worked solution are below.

TMUA 2023 · Paper 1Coordinate geometryInscribed rectangle · quadratic in a length6 options

The original question includes a diagram: A square MNOP with a rectangle RSTU inscribed at an angle, one vertex on each edge of the square.

The following shape has two lines of reflectional symmetry.

MNOP is a square of perimeter 40 cm. The vertices of rectangle RSTU lie on the edges of square MNOP, with R on MN, S on NO, T on OP and U on MP.

MR has length x cm.

What is the largest possible value of x such that RSTU has area 20 cm²?

  1. A√2
  2. B√10
  3. C2√15
  4. D10√2
  5. E5 + √5
  6. F5 + √15
Show the answer and worked solution
answer · F
  1. A√2
  2. B√10
  3. C2√15
  4. D10√2
  5. E5 + √5
  6. F5 + √15
The square has side 10. Put M at the origin with the square in the first quadrant. The symmetry forces MU = MR = x, so R = (0,  x), U = (x,  0), T = (10,  10−x) and S = (10−x,  10). Then RU has length x√2 and UT has length (10−x)√2, so the area is 2x(10−x). Setting 2x(10−x) = 20 gives x2 − 10x + 10 = 0, so x = 5 ± √15. The larger is 5 + √15.