Right-angled Triangles

Right-angled Triangles, SM-Bank 053

Use Pythagoras' Theorem to calculate the length of the hypotenuse in the isosceles triangle below. Give your answer correct in exact surd form. (2 marks)

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Using your result from part (a) above, calculate the perimeter of the shape below, correct to 1 decimal place. (2 marks)

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a. \(\sqrt{50}\ \text{cm (exact surd form)}\)

b. \(30.6\ \text{cm (1 d.p.)}\)

Show Worked Solution

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

\(\text{Let }a=5\ \text{and }b=5\)

\(\text{Then}\ \ c^2\)	\(=5^2+5^2\)
\(c^2\)	\(=50\)
\(c\)	\(=\sqrt{50}\ \text{cm (exact surd form)}\)

b.	\(\text{Perimeter}\)	\(=\text{chord (a)}\ +\dfrac{3}{4}\times\text{circumference}\)
		\(=\sqrt{50}+\dfrac{3}{4}\times 2\pi r\)
		\(=\sqrt{50}+\dfrac{3}{4}\times 2\pi\times 5\)
		\(=30.633\dots\)
		\(\approx 30.6\ \text{cm (1 d.p.)}\)

Right-angled Triangles, SM-Bank 052

Use Pythagoras' Theorem to calculate the perimeter of the isosceles triangle below, correct to the nearest centimetre. (3 marks)

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\(29\ \text{cm}\ (\text{nearest cm})\)

Show Worked Solution

\(\text{Find length of the equal sides of the triangle.}\)

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

\(\text{Let }a=5\ \text{and }b=8\)

\(\text{Then}\ \ c^2\)	\(=5^2+8^2\)
\(c^2\)	\(=89\)
\(c\)	\(=\sqrt{89}\)
\(c\)	\(=9.433\dots\)

\(\text{Perimeter}\)

\(=2\times 9.433\dots+10\)

\(=28.867\dots\approx 29\ \text{cm}\ (\text{nearest cm})\)

Right-angled Triangles, SM-Bank 051

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

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\(31.1\ \text{mm}\ (1\ \text{d.p.})\)

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\(\text{Find length of sloped side of trapezium.}\)

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

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

\(\text{Then}\ \ c^2\)	\(=4^2+7^2\)
\(c^2\)	\(=65\)
\(c\)	\(=\sqrt{65}\)
\(c\)	\(=8.062\dots\)

\(\text{Perimeter}\)

\(=6+7+10+8.062\dots\)

\(=31.062\dots\approx 31.1\ \text{mm}\ (1\ \text{d.p.})\)

Right-angled Triangles, SM-Bank 050

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

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

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

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

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

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

\(\text{Perimeter}\)

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

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

Right-angled Triangles, SM-Bank 049

Calculate the length of the hypotenuse in the following right-angled triangle. Give your answer correct to the nearest metre. (2 marks)
Find the perimeter of the triangle. (1 mark)

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a. \(7\ \text{m}\)

b. \(18\ \text{m}\)

Show Worked Solution

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

\(\text{Let }a=5\ \text{and }b=6\)

\(\text{Then}\ \ c^2\)	\(=5^2+6^2\)
\(c^2\)	\(=61\)
\(c\)	\(=\sqrt{61}\)
\(c\)	\(=7.141\dots\approx 7\)

\(\text{The hypotenuse is }7\ \text{metres (nearest metre})\)

b. \(\text{Perimeter}\)

\(=7+5+6\)

\(=18\ \text{m}\)

Right-angled Triangles, SM-Bank 048

The size of a computer monitor is determined by the length of the diagonal of the screen. Use Pythagoras' Theorem to calculate the size of the monitor in the diagram below. (2 marks)

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\(51\ \text{centimetres}\)

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

\(\text{Let }a=24\ \text{and }b=45\)

\(\text{Then}\ \ c^2\)	\(=24^2+45^2\)
\(c^2\)	\(=2601\)
\(c\)	\(=\sqrt{2601}\)
\(c\)	\(=51\)

\(\text{The size of the monitor is }51\ \text{centimetres}\)

Right-angled Triangles, SM-Bank 047

Use Pythagoras' Theorem to calculate the length of the diagonal in the rectangle below, correct to 1 decimal place. (2 marks)

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\(8.4\ \text{centimetres}\ (1\ \text{d.p.})\)

Show Worked Solution

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

\(\text{Let }a=4.2\ \text{and }b=7.3\)

\(\text{Then}\ \ c^2\)	\(=4.2^2+7.3^2\)
\(c^2\)	\(=70.98\)
\(c\)	\(=\sqrt{70.98}\)
\(c\)	\(=8.4219\dots\approx 8.4\ (1\ \text{d.p.})\)

\(\text{The length of the string is }8.4\ \text{centimetres}\ (1\ \text{d.p.})\)

Right-angled Triangles, SM-Bank 046

Lewis is holding a string attached to a kite. Lewis is 8 metres from the kite and the kite is flying 4 metres above his hand. Use Pythagoras' Theorem to calculate the length of the string attached to the kite, correct to 2 decimal places. (2 marks)

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\(8.94\ \text{metres}\ (2\ \text{d.p.})\)

Show Worked Solution

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

\(\text{Let }a=4\ \text{and }b=8\)

\(\text{Then}\ \ c^2\)	\(=4^2+8^2\)
\(c^2\)	\(=80\)
\(c\)	\(=\sqrt{80}\)
\(c\)	\(=8.9442\dots\approx 8.94\ (2\ \text{d.p.})\)

\(\text{The length of the string is }8.94\ \text{metres}\ (2\ \text{d.p.})\)

Right-angled Triangles, SM-Bank 045

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

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\(80\ \text{metres}\)

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)

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

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 042

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

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\(c=\sqrt{277}\)

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

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

\(\text{Then}\ \ c^2\)	\(=9^2+14^2\)
\(c^2\)	\(=277\)
\(c\)	\(=\sqrt{277}\)

Right-angled Triangles, SM-Bank 041

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

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\(c=\sqrt{269}\)

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

\(\text{Let }a=10,\ \text{and }b=13\)

\(\text{Then}\ \ c^2\)	\(=10^2+13^2\)
\(c^2\)	\(=269\)
\(c\)	\(=\sqrt{269}\)

Right-angled Triangles, SM-Bank 040

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

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\(c=\sqrt{98}\)

Show Worked Solution

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

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

\(\text{Then}\ \ c^2\)	\(=7^2+7^2\)
\(c^2\)	\(=98\)
\(c\)	\(=\sqrt{98}\)

Right-angled Triangles, SM-Bank 039

Find the value of the pronumeral in the right-angled triangle below, giving your answer correct to 1 decimal place. (2 marks)

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\(c\approx 6.5\ (1\ \text{d.p.})\)

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

\(\text{Let }a=4.8,\ \text{and }b=4.4\)

\(\text{Then}\ \ c^2\)	\(=4.8^2+4.4^2\)
\(c^2\)	\(=42.4\)
\(c\)	\(=\sqrt{42.4}\)
\(c\)	\(=6.511\dots\)
\(c\)	\(\approx 6.5\ (1\ \text{d.p.})\)

Right-angled Triangles, SM-Bank 038

Find the value of the pronumeral in the right-angled triangle below, giving your answer correct to 1 decimal place. (2 marks)

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\(c\approx 16.2\ (1\ \text{d.p.})\)

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

\(\text{Let }a=15.4,\ \text{and }b=5.1\)

\(\text{Then}\ \ c^2\)	\(=15.4^2+5.1^2\)
\(c^2\)	\(=263.17\)
\(c\)	\(=\sqrt{263.17}\)
\(c\)	\(=16.222\dots\)
\(c\)	\(\approx 16.2\ (1\ \text{d.p.})\)

Right-angled Triangles, SM-Bank 037

Find the value of the pronumeral in the right-angled triangle below, giving your answer correct to 1 decimal place. (2 marks)

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\(c\approx 12.5\ (1\ \text{d.p.})\)

Show Worked Solution

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

\(\text{Let }a=10.4,\ \text{and }b=6.9\)

\(\text{Then}\ \ c^2\)	\(=10.4^2+6.9^2\)
\(c^2\)	\(=155.77\)
\(c\)	\(=\sqrt{155.77}\)
\(c\)	\(=12.480\dots\)
\(c\)	\(\approx 12.5\ (1\ \text{d.p.})\)

Right-angled Triangles, SM-Bank 036

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

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\(c=70\)

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

\(\text{Let }a=42,\ \text{and }b=56\)

\(\text{Then}\ \ c^2\)	\(=42^2+56^2\)
\(c^2\)	\(=4900\)
\(c\)	\(=\sqrt{4900}\)
\(c\)	\(=70\)

Right-angled Triangles, SM-Bank 035

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

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\(c=15\)

Show Worked Solution

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

\(\text{Let }a=9,\ \text{and }b=12\)

\(\text{Then}\ \ c^2\)	\(=9^2+12^2\)
\(c^2\)	\(=225\)
\(c\)	\(=\sqrt{225}\)
\(c\)	\(=15\)

Right-angled Triangles, SM-Bank 034

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

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\(x=25\)

Show Worked Solution

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

\(\text{Let }a=24,\ b=7\ \text{and }c=x\)

\(\text{Then}\ \ x^2\)	\(=24^2+7^2\)
\(x^2\)	\(=625\)
\(x\)	\(=\sqrt{625}\)
\(x\)	\(=25\)

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)

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\(5590\ \text{m}\)

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)

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\(a=\sqrt{45}\)

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)

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\(a=\sqrt{176}\)

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)

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\(a=\sqrt{115}\)

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)

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

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)

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

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)

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

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)

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\(x=12\)

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)

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\(x=24\)

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)

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\(4.8\ \text{m}\)

Show Worked Solution

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

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

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

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

Right-angled Triangles, SM-Bank 024

Use Pythagoras' Theorem to decide if the triangle below is right-angled. (2 marks)

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\(\text{See worked solution}\)

\(\text{The triangle is not right-angled as the sides do not form a Pythagorean triad.}\)

Show Worked Solution

\(\text{Let }a=19 ,\ b=23\ \text{and }c=24\)

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

\(\text{LHS: }\rightarrow\ \)	\(a^2+b^2\)	\(=19^2+23^2\)
		\(=361+529\)
		\(=890\)
		\(\ne 24^2\)
	\(\therefore\ \text{LHS}\)	\(\ne \text{RHS}\)

\(\therefore\ \text{The triangle is not right-angled as the sides do not form a Pythagorean triad.}\)

Right-angled Triangles, SM-Bank 023

Use Pythagoras' Theorem to decide if the triangle below is right-angled. (2 marks)

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\(\text{See worked solution}\)

\(\text{The triangle is not right-angled as the sides do not form a Pythagorean triad.}\)

Show Worked Solution

\(\text{Let }a=7 ,\ b=15\ \text{and }c=17\)

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

\(\text{LHS: }\rightarrow\ \)	\(a^2+b^2\)	\(=7^2+15^2\)
		\(=49+225\)
		\(=274\)
		\(\ne 17^2\)
	\(\therefore\ \text{LHS}\)	\(\ne \text{RHS}\)

\(\therefore\ \text{The triangle is not right-angled as the sides do not form a Pythagorean triad.}\)

Right-angled Triangles, SM-Bank 022

Use Pythagoras' Theorem to decide if the triangle below is right-angled. (2 marks)

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\(\text{See worked solution}\)

\(\text{The triangle is not right-angled as the sides do not form a Pythagorean triad.}\)

Show Worked Solution

\(\text{Let }a=6 ,\ b=7\ \text{and }c=8\)

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

\(\text{LHS: }\rightarrow\ \)	\(a^2+b^2\)	\(=6^2+7^2\)
		\(=36+49\)
		\(=85\)
		\(\ne 8^2\)
	\(\therefore\ \text{LHS}\)	\(\ne \text{RHS}\)

\(\therefore\ \text{The triangle is not right-angled as the sides do not form a Pythagorean triad.}\)

Right-angled Triangles, SM-Bank 021

Use Pythagoras' Theorem to decide if the triangle below is right-angled. (2 marks)

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\(\text{See worked solution}\)

\(\text{The triangle is right-angled as the sides form a Pythagorean triad.}\)

Show Worked Solution

\(\text{Let }a=13 ,\ b=84\ \text{and }c=85\)

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

\(\text{LHS: }\rightarrow\ \)	\(a^2+b^2\)	\(=13^2+84^2\)
		\(=169+7056\)
		\(=7225\)
		\(=85^2\)
		\(=\text{RHS}\)

\(\therefore\ \text{The triangle is right-angled as the sides form a Pythagorean triad.}\)

Right-angled Triangles, SM-Bank 020

Use Pythagoras' Theorem to decide if the triangle below is right-angled. (2 marks)

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\(\text{See worked solution}\)

\(\text{The triangle is right-angled as the sides form a Pythagorean triad.}\)

Show Worked Solution

\(\text{Let }a=1.5 ,\ b=2\ \text{and }c=2.5\)

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

\(\text{LHS: }\rightarrow\ \)	\(a^2+b^2\)	\(=1.5^2+2^2\)
		\(=2.25+4\)
		\(=6.25\)
		\(=2.5^2\)
		\(=\text{RHS}\)

\(\therefore\ \text{The triangle is right-angled as the sides form a Pythagorean triad.}\)

Right-angled Triangles, SM-Bank 019

Use Pythagoras' Theorem to decide if the triangle below is right-angled. (2 marks)

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\(\text{See worked solution}\)

\(\text{The triangle is right-angled as the sides form a Pythagorean triad.}\)

Show Worked Solution

\(\text{Let }a=24 ,\ b=7\ \text{and }c=25\)

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

\(\text{LHS: }\rightarrow\ \)	\(a^2+b^2\)	\(=24^2+7^2\)
		\(=576+49\)
		\(=625\)
		\(=25^2\)
		\(=\text{RHS}\)

\(\therefore\ \text{The triangle is right-angled as the sides form a Pythagorean triad.}\)

Right-angled Triangles, SM-Bank 018

Use Pythagoras' Theorem to decide if the numbers \(7, 8\) and \(11\) form a Pythagorean triad. (2 marks)

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\(\text{See worked solution}\)

\(7,\ 8\ \text{and }11\ \text{do not form a Pythagorean triad.}\)

Show Worked Solution

\(\text{Let }a=7 ,\ b=8\ \text{and }c=11\)

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

\(\text{LHS: }\rightarrow\ \)	\(a^2+b^2\)	\(=7^2+8^2\)
		\(=49+64\)
		\(=113\)
		\(\ne 11^2\)
	\(\therefore\ \ \)	\(\ne\text{RHS}\)

\(\therefore\ 7,\ 8\ \text{and }11\ \text{do not form a Pythagorean triad.}\)

Right-angled Triangles, SM-Bank 017

Use Pythagoras' Theorem to decide if the numbers \(1.4, 4.8\) and \(5.2\) form a Pythagorean triad. (2 marks)

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\(\text{See worked solution}\)

\(1.4,\ 4.8\ \text{and }5.2\ \text{do not form a Pythagorean triad.}\)

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\(\text{Let }a=1.4 ,\ b=4.8\ \text{and }c=5.2\)

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

\(\text{LHS: }\rightarrow\ \)	\(a^2+b^2\)	\(=1.4^2+4.8^2\)
		\(=1.96+23.04\)
		\(=25\)
		\(=5^2\ne\text{RHS}\)

\(\therefore\ 1.4,\ 4.8\ \text{and }5.2\ \text{do not form a Pythagorean triad.}\)

Right-angled Triangles, SM-Bank 016

Use Pythagoras' Theorem to decide if the numbers \(36, 48\) and \(63\) form a Pythagorean triad. (2 marks)

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\(\text{See worked solution}\)

\(36,\ 48\ \text{and }63\ \text{do not form a Pythagorean triad.}\)

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\(\text{Let }a=36 ,\ b=48\ \text{and }c=63\)

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

\(\text{LHS: }\longrightarrow\ \)	\(a^2+b^2\)	\(=36^2+48^2\)
		\(=1296+2304\)
		\(=3600\)
		\(=60^2\ne\text{RHS}\)

\(\therefore\ 36,\ 48\ \text{and }63\ \text{do not form a Pythagorean triad.}\)

Right-angled Triangles, SM-Bank 015

Use Pythagoras' Theorem to decide if the numbers \(1.1 ,\ 6\) and \(6.1\) form a Pythagorean triad. (2 marks)

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\(\text{See worked solution}\)

\(1.1,\ 6\ \text{and }6.1\ \text{form a Pythagorean triad.}\)

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\(\text{Let }a=1.1 ,\ b=6\ \text{and }c=6.1\)

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

\(\text{LHS: }\longrightarrow\ \)	\(a^2+b^2\)	\(=1.1^2+6^2\)
		\(=1.21+36\)
		\(=37.21\)
		\(=6.1^2=\text{RHS}\)

\(\therefore\ 1.1,\ 6\ \text{and }6.1\ \text{form a Pythagorean triad.}\)

Right-angled Triangles, SM-Bank 014

Use Pythagoras' Theorem to decide if the numbers \(12 , 35\) and \(37\) form a Pythagorean triad. (2 marks)

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\(\text{See worked solution}\)

\(12,\ 35\ \text{and }37\ \text{form a Pythagorean triad.}\)

Show Worked Solution

\(\text{Let }a=12 ,\ b=35\ \text{and }c=37\)

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

\(\text{LHS: }\longrightarrow\ \)	\(a^2+b^2\)	\(=12^2+35^2\)
		\(=144+1225\)
		\(=1369\)
		\(=37^2=\text{RHS}\)

\(\therefore\ 12,\ 35\ \text{and }37\ \text{form a Pythagorean triad.}\)

Right-angled Triangles, SM-Bank 013

Use Pythagoras' Theorem to decide if the numbers \(5 , 12\) and \(13\) form a Pythagorean triad. (2 marks)

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\(\text{See worked solution}\)

Show Worked Solution

\(\text{Let }a=5 ,\ b=12\ \text{and }c=13\)

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

\(\text{LHS: }\longrightarrow\ \)	\(a^2+b^2\)	\(=5^2+12^2\)
		\(=25+144\)
		\(=169\)
		\(=13^2=\text{RHS}\)

\(\therefore\ 5,\ 12\ \text{and }13\ \text{form a Pythagorean triad.}\)

Right-angled Triangles, SM-Bank 012

Which side is the hypotenuse in these right angled triangles? (3 marks)

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a. \(w\)

b. \(q\)

c. \(e\)

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a. \(\text{Side: }w\)

b. \(\text{Side: }q\)

c. \(\text{Side: }e\)

Right-angled Triangles, SM-Bank 011

Which side is the hypotenuse in these right angled triangles? (3 marks)

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a. \(x\)

b. \(m\)

c. \(a\)

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a. \(\text{Side: }x\)

b. \(\text{Side: }m\)

c. \(\text{Side: }a\)

Right-angled Triangles, SM-Bank 010 MC

Which of the following is true of the hypotenuse in a right-angled triangle?

The hypotenuse is the side opposite to the right angle.
The hypotenuse is the side adjacent to the right angle.
The hypotenuse is one of the shorter sides in a right angled triangle.
The hypotenuse is equal to the sum of the square of the other two sides.

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\(A\)

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\(\text{The hypotenuse is the side opposite to the right angle.}\)

\(\Rightarrow A\)

Right-angled Triangles, SM-Bank 009 MC

Which of the following correctly states Pythagoras' Theorem for this triangle?

\(b^2=c^2-a^2\)
\(c^2=b^2+c^2\)
\(a^2=b^2-c^2\)
\(c^2=a^2-b^2\)

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\(C\)

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\(a^2=b^2-c^2\)

\(\Rightarrow C\)

Right-angled Triangles, SM-Bank 008 MC

Which of the following correctly states Pythagoras' Theorem for this triangle?

\(w^2=x^2+z^2\)
\(z^2=w^2-x^2\)
\(x^2=z^2-w^2\)
\(w^2=x^2-z^2\)

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\(D\)

Show Worked Solution

\(w^2=x^2-z^2\)

\(\Rightarrow D\)

Right-angled Triangles, SM-Bank 007 MC

Which of the following correctly states Pythagoras' Theorem for this triangle?

\(c^2=a^2-b^2\)
\(b^2=c^2-a^2\)
\(a^2=b^2-c^2\)
\(c^2=a^2+b^2\)

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\(A\)

Show Worked Solution

\(c^2=a^2-b^2\)

\(\Rightarrow A\)

Right-angled Triangles, SM-Bank 006 MC

Which of the following correctly states Pythagoras' Theorem for this triangle?

\(m^2=n^2-p^2\)
\(p^2=m^2-n^2\)
\(n^2=p^2-m^2\)
\(n^2=m^2+p^2\)

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\(B\)

Show Worked Solution

\(p^2=m^2-n^2\)

\(\Rightarrow B\)

Right-angled Triangles, SM-Bank 005 MC

Which of the following correctly states Pythagoras' Theorem for this triangle?

\(m^2=n^2+p^2\)
\(p^2=m^2+n^2\)
\(n^2=p^2+m^2\)
\(n^2=m^2-p^2\)

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\(A\)

Show Worked Solution

\(m^2=n^2+p^2\)

\(\Rightarrow A\)

Right-angled Triangles, SM-Bank 004 MC

Which of the following correctly states Pythagoras' Theorem for this triangle?

\(w^2=x^2+z^2\)
\(x^2=w^2+z^2\)
\(z^2=x^2+w^2\)
\(x^2=w^2-z^2\)

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\(B\)

Show Worked Solution

\(x^2=w^2+z^2\)

\(\Rightarrow B\)

Right-angled Triangles, SM-Bank 003 MC

Which of the following correctly states Pythagoras' Theorem for this triangle?

\(c^2=a^2+b^2\)
\(b^2=c^2-a^2\)
\(a^2=b^2+c^2\)
\(b^2=a^2+c^2\)

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\(C\)

Show Worked Solution

\(a^2=b^2+c^2\)

\(\Rightarrow C\)

Right-angled Triangles, SM-Bank 002 MC

Which of the following correctly states Pythagoras' Theorem for this triangle?

\(c^2=a^2+b^2\)
\(b^2=c^2-a^2\)
\(a^2=b^2+c^2\)
\(b^2=a^2+c^2\)

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\(D\)

Show Worked Solution

\(b^2=a^2+c^2\)

\(\Rightarrow D\)

Right-angled Triangles, SM-Bank 001 MC

Henry flies a kite attached to a long string, as shown in the diagram below.

The horizontal distance of the kite to Henry’s hand is 8 m.

The vertical distance of the kite above Henry’s hand is 15 m.

The length of the string, in metres, is

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\(B\)

Show Worked Solution

\(\text{Using Pythagoras}\)

\(s^2\)	\(= 8^2 + 15^2\)
	\(= 289\)
\(\therefore\ s\)	\(=\sqrt{289}\)
	\(= 17\ \text{metres}\)

\(\Rightarrow B\)

Measurement, STD2 M6 2011 HSC 9 MC

Two trees on level ground, 12 metres apart, are joined by a cable. It is attached 2 metres above the ground to one tree and 11 metres above the ground to the other.

What is the length of the cable between the two trees, correct to the nearest metre?

`9\ text(m)`
`12\ text(m)`
`15\ text(m)`
`16\ text(m)`

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`C`

Show Worked Solution

`text(Using Pythagoras)`

`c^2`	`=12^2+9^2`
	`=144+81`
	`=225`
`:.c`	`=15,\ \ c>0`

`=>C`