Converse of Pythagoras theoremIB MYP Maths Extended: Subtopic test

1. 1
   A triangle has sides of length $9$ cm, $12$ cm and $15$ cm.

   (a)

   Which calculation shows that the triangle is right-angled?

   [1 mark]

   • A$9^2+15^2=12^2$
   • B$9+12=15$
   • C$12^2+15^2=9^2$
   • D$9^2+12^2=15^2$

   (b)

   Opposite which side is the right angle?

   [1 mark]

   • Athe side of length 9 cm
   • Bthe side of length 12 cm
   • Cthe side of length 15 cm
   • Dnone of the sides

   (c)

   Find the area of the triangle.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Carpenters check that the corner of a rectangular frame is square by measuring the two sides at the corner and the diagonal opposite it. For one frame the sides are $2.4$ m and $1.8$ m, and the diagonal is measured as $3.1$ m.

   (a)

   Find the value of $2.4^2+1.8^2$.

   [1 mark]

   • A9
   • B4.2
   • C9.61
   • D3

   (b)

   What does the measurement $3.1$ m tell the carpenter about the corner between the $2.4$ m and $1.8$ m sides?

   [1 mark]

   • AIt is exactly $90^\circ$
   • BIt is greater than $90^\circ$
   • CIt is less than $90^\circ$
   • DIt cannot be decided from these measurements

   (c)

   The carpenter adjusts the frame until the diagonal measures exactly $3.0$ m. Explain, with a calculation, why the corner is now a right angle.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Aisha investigates whole-number triples $(a,b,c)$ that satisfy $a^2+b^2=c^2$, called Pythagorean triples. She finds $(3,4,5)$, $(6,8,10)$ and $(9,12,15)$.

   (a)

   Describe the pattern in the three triples and write a general rule for further triples. Verify your rule using the next triple in the pattern.

   [3 marks]

   (b)

   Aisha then tests triangles with side lengths (i) $7$, $24$, $25$ (ii) $8$, $15$, $16$ and (iii) $5$, $6$, $9$ (all in cm). For each triangle, decide whether it is acute-angled, right-angled or obtuse-angled. Show your working.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A landscape architect measures a triangular vegetable plot. The three sides measure $6.1$ m, $8.0$ m and $10.1$ m, each measured to the nearest $0.1$ m. The architect wants a right-angled corner between the two shorter sides, so that a rectangular greenhouse can fit against them.

   (a)

   Use the measurements to classify the plot by its angles (acute, right-angled or obtuse) and by its sides (equilateral, isosceles or scalene). Show your working.

   [6 marks]

   (b)

   Justify whether the architect can be sure that the corner between the two shorter sides is not a right angle.

   [6 marks]

   Total for question 4: 12 marks

End of questions