SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Right-angled Triangles, SM-Bank 050

Use Pythagoras' Theorem to calculate the perimeter of the rectangle below, correct to 1 decimal place.  (3 marks)

Show Answers Only

\(13.0\ \text{m}\ (1\ \text{d.p.})\)

Show Worked Solution

\(\text{Find side length of rectangle.}\)

\(\text{Pythagoras’ Theorem states:  }a^2+b^2=c^2\)

\(\text{Let }b=3\ \text{and }c=4.6\)

\(\text{Then}\ \ a^2+3^2\) \(=4.6^2\)
\(a^2\) \(=4.6^2-3^2\)
\(a\) \(=\sqrt{12.16}\)
\(a\) \(=3.487\dots\)

 
\(\text{Perimeter}\)

\(=2\times 3.487\dots+ 2\times 3\)

\(=12.974\dots\approx 13.0\ \text{m}\ (1\ \text{d.p.})\)

Right-angled Triangles, SM-Bank 045

Use Pythagoras' Theorem to calculate the height of the building below.  (2 marks)

Show Answers Only

\(80\ \text{metres}\)

Show Worked Solution

\(\text{Pythagoras’ Theorem states:  }a^2+b^2=c^2\)

\(\text{Let }a=h,\ b=150\ \text{and }c=170\)

\(h^2+150^2\) \(=170^2\)
\(h^2\) \(=170^2-150^2\)
\(h^2\) \(=6400\)
\(h\) \(=\sqrt{6400}\)
\(h\) \(=80\)

 
\(\text{The height of the building is }80\ \text{metres}.\)

Right-angled Triangles, SM-Bank 044

Calculate the perpendicular height of the isosceles triangle below, giving your answer correct to one decimal place.  (2 marks)

 

Show Answers Only

\(\approx 6.9\ (1\text{ d.p.})\)

Show Worked Solution

\(\text{Pythagoras’ Theorem states:  }a^2+b^2=c^2\)

\(\text{Let }b=3\ \text{and }c=7.5\)

\(a^2+3^2\) \(=7.5^2\)
\(a^2\) \(=7.5^2-3^2\)
\(a^2\) \(=47.25\)
\(a\) \(=\sqrt{46.01}\)
\(a\) \(=6.873\dots\approx 6.9\ (1\text{ d.p.})\)

Right-angled Triangles, SM-Bank 043

Arthur observes a plane through his binoculars at a distance of 7500 metres. If Ben is directly under the plane and it is flying at an altitude of 5000 metres, how far apart are Arthur and Ben?  Give your answer correct to the nearest metre.  (2 Marks)

 

Show Answers Only

\(5590\ \text{m}\)

Show Worked Solution

\(\text{Using Pythagoras’ Theorem: }\ \ a^2+b^2=c^2\)

\(\text{Let }\ a=d,\  b=5000,\  c=7500\)

\(d^2+5000^2\) \(=7500^2\)
\(d^2\) \(=7500^2-5000^2\)
\(d^2\) \(=31\ 250\ 000\)
\(d\) \(=5590\dots\)
\(d\) \(\approx 5590\)

 

\(\therefore\ \text{Arthur and Ben are approximately }5590\ \text{metres apart}.\)

Right-angled Triangles, SM-Bank 033

Find the value of the pronumeral in the right-angled triangle below giving your answer in exact surd form.  (2 marks)

Show Answers Only

\(a=\sqrt{45}\)

Show Worked Solution

\(\text{Pythagoras’ Theorem states:  }a^2+b^2=c^2\)

\(\text{Let }b=6\ \text{and }c=9\)

\(a^2+6^2\) \(=9^2\)
\(a^2+36\) \(=81\)
\(a^2\) \(=81-36\)
\(a^2\) \(=45\)
\(\therefore a\) \(=\sqrt{45}\)

Right-angled Triangles, SM-Bank 032

Find the value of the pronumeral in the right-angled triangle below giving your answer in exact surd form.  (2 marks)

Show Answers Only

\(a=\sqrt{176}\)

Show Worked Solution

\(\text{Pythagoras’ Theorem states:  }a^2+b^2=c^2\)

\(\text{Let }b=20\ \text{and }c=24\)

\(a^2+20^2\) \(=24^2\)
\(a^2+400\) \(=576\)
\(a^2\) \(=576-400\)
\(a^2\) \(=176\)
\(\therefore a\) \(=\sqrt{176}\)

Right-angled Triangles, SM-Bank 031

Find the value of the pronumeral in the right-angled triangle below giving your answer in exact surd form.  (2 marks)

Show Answers Only

\(a=\sqrt{115}\)

Show Worked Solution

\(\text{Pythagoras’ Theorem states:  }a^2+b^2=c^2\)

\(\text{Let }b=9\ \text{and }c=14\)

\(a^2+9^2\) \(=14^2\)
\(a^2+81\) \(=196\)
\(a^2\) \(=196-81\)
\(a^2\) \(=115\)
\(\therefore a\) \(=\sqrt{115}\)

Right-angled Triangles, SM-Bank 030

Find the value of the pronumeral in the right-angled triangle below correct to one decimal place.  (2 marks)

Show Answers Only

\(a=28.3\ \text{(1 d.p.)}\)

Show Worked Solution

\(\text{Pythagoras’ Theorem states:  }a^2+b^2=c^2\)

\(\text{Let }b=98\ \text{and }c=102\)

\(a^2+98^2\) \(=102^2\)
\(a^2+9604\) \(=10404\)
\(a^2\) \(=10404-9604\)
\(a^2\) \(=800\)
\(\therefore a\) \(=\sqrt{800}\)
  \(=28.284\dots\approx 28.3\ \text{(1 d.p.)}\)

Right-angled Triangles, SM-Bank 029

Find the value of the pronumeral in the right-angled triangle below correct to one decimal place.  (2 marks)

Show Answers Only

\(a=15.7\ \text{(1 d.p.)}\)

Show Worked Solution

\(\text{Pythagoras’ Theorem states:  }a^2+b^2=c^2\)

\(\text{Let }b=14\ \text{and }c=21\)

\(a^2+14^2\) \(=21^2\)
\(a^2+196\) \(=441\)
\(a^2\) \(=441-196\)
\(a^2\) \(=245\)
\(\therefore a\) \(=\sqrt{245}\)
  \(=15.652\dots\approx 15.7\ \text{(1 d.p.)}\)

Right-angled Triangles, SM-Bank 028

Find the value of the pronumeral in the right-angled triangle below correct to one decimal place.  (2 marks)

Show Answers Only

\(x=5.7\ \text{(1 d.p.)}\)

Show Worked Solution

\(\text{Pythagoras’ Theorem states:  }a^2+b^2=c^2\)

\(\text{Let }a=x ,\ b=4\ \text{and }c=7\)

\(x^2+4^2\) \(=7^2\)
\(x^2+16\) \(=49\)
\(x^2\) \(=49-16\)
\(x^2\) \(=33\)
\(\therefore x\) \(=\sqrt{33}\)
  \(=5.744\dots\approx 5.7\ \text{(1 d.p.)}\)

Right-angled Triangles, SM-Bank 027

Find the value of the pronumeral in the right-angled triangle below.  (2 marks)

Show Answers Only

\(x=12\)

Show Worked Solution

\(\text{Pythagoras’ Theorem states:  }a^2+b^2=c^2\)

\(\text{Let }a=x ,\ b=7\ \text{and }c=25\)

\(x^2+9^2\) \(=15^2\)
\(x^2+81\) \(=225\)
\(x^2\) \(=225-81\)
\(x^2\) \(=144\)
\(\therefore x\) \(=\sqrt{144}\)
  \(=12\)

Right-angled Triangles, SM-Bank 026

Find the value of the pronumeral in the right-angled triangle below.  (2 marks)

Show Answers Only

\(x=24\)

Show Worked Solution

\(\text{Pythagoras’ Theorem states:  }a^2+b^2=c^2\)

\(\text{Let }a=x ,\ b=7\ \text{and }c=25\)

\(x^2+7^2\) \(=25^2\)
\(x^2+49\) \(=625\)
\(x^2\) \(=625-49\)
\(x^2\) \(=576\)
\(\therefore x\) \(=\sqrt{576}\)
  \(=24\)

Right-angled Triangles, SM-Bank 025

Find the height of the power pole below correct to one decimal place.  (2 marks)

Show Answers Only

\(4.8\ \text{m}\)

Show Worked Solution

\(\text{Let }a=x ,\ b=3.6\ \text{and }c=6\)

\(\text{Pythagoras’ Theorem states:  }a^2+b^2=c^2\)

\(x^2+3.6^2\) \(=6^2\)
\(x^2+12.96\) \(=36\)
\(x^2\) \(=36-12.96\)
\(x^2\) \(=23.04\)
\(\therefore x\) \(=\sqrt{23.04}\)
  \(=4.8\)

\(\therefore\ \text{The height of the power pole is }4.8\ \text{m}\)