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This investigation looks at finding the integer lengths of the sides of a cuboid that has an integer length for its diagonal.
The diagram shows a cuboid with sides of length 3, 5 and 6.
(a) Using Pythagoras’ Theorem in triangle $PQR$ gives $PR^2 = 3^2 + 5^2$.
Find the value of $PR^2$.
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(b) Using Pythagoras’ Theorem in triangle $PRS$ gives $PS^2 = PR^2 + 6^2$.
Find the value of $PS^2$.
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578
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533
The diagram shows a cuboid with sides of integer length $a$, $b$ and $c$. Its diagonal, $PS$, has integer length $d$.
(a) Use Pythagoras' Theorem in triangle $PQR$ to write down an expression for $PR^2$ in terms of $a$ and $b$. .............................................. [1]
(b) Use your answer to part (a), and Pythagoras' Theorem in triangle $PRS$, to show that $$d^2 = a^2 + b^2 + c^2.$$ [1]
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A cuboid has sides of length $a$, $b$ and $c$, where $a$, $b$ and $c$ are integers and $a \leq b \leq c$. If the length of the diagonal, $d$, is also an integer then $(a, b, c, d)$ is a Pythagorean set of four.
Use $d^2 = a^2 + b^2 + c^2$ to show that a cuboid with sides of length 4, 17 and 28 gives a Pythagorean set of four. Complete the Pythagorean set of four.
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(a) In a Pythagorean set of four $(a, b, c, d)$ $d^2 = a^2 + b^2 + c^2$.
When $d = a + c$, show that $ac = \frac{b^2}{2}$. [2]
(b) Explain why $b$ must be even.
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Here is a method for finding Pythagorean sets of four using Question 4:
- Choose any even integer for $b$.
- Calculate $ac$ using Question 4(a).
- Find all the possible pairs of integers for $a$ and $c$, where $a < c$.
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(a) Which one of these two sets is a Pythagorean set of four?
(18, 24, 72, 78) or (18, 24, 72, 84)?
Show how you decide.
[2]
(b) $(ka, kb, kc, kd)$ is a Pythagorean set of four, where $k$ is a positive integer greater than 1.
Use algebra to show that $(a, b, c, d)$ must also be a Pythagorean set of four.
[2]
(c) $(a, b, c, d)$ is a basic Pythagorean set of four if the numbers $a, b, c$ and $d$ have no common factor greater than 1.
Find the basic Pythagorean set of four for your answer to part (a).
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592
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The method in Question 5 to find Pythagorean sets of four is:
• Choose any even integer for $b$.
• Calculate $ac$ using Question 4(a).
• Find all the possible pairs of integers for $a$ and $c$, where $a < c$.
Use this method to find two basic Pythagorean sets of four where the smallest integer, $a$, is 2.
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This task looks at models for the height of the image of a reflected laser beam on a vertical wall. In this task all the measurements are in metres.
The diagram shows, by a dashed line, the side view of the path of a laser beam.
The laser beam
• starts at source $L$
• travels to a point $R$ on horizontal ground $AB$
• reflects at the point $R$ so that angle $LRA = \text{angle } NRB$
• travels to $N$, its image, on a vertical wall.
The height of $L$ above the horizontal ground is $LA = 4$.
The height of $N$ above the horizontal ground is $NB = h$.
$AR = 6$ and $RB = 15$.
Complete the statements to show that $h = 10$.
Triangle $LRA$ is similar to triangle $NRB$.
$\frac{h}{4} = \frac{15}{\ldots}\h = \frac{\ldots \times \ldots}{\ldots} = 10$
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The laser source, L, can move towards or away from the wall. It now moves x metres to the right so that AR becomes $6 - x$ metres. The point R does not move. The other given measurements remain the same.
When the laser beam reflects at R, triangle LRA and triangle NRB will always be similar.
!(Image URL)
(a) Use the method in Question 8 to find a model for h in terms of x. ......................................................... [2]
(b) Sketch the graph of h against x for $-6 < x < 6$. ......................................................... [2]
(c) (i) Write down the equation of the vertical asymptote. ......................................................... [2]
(ii) Give a reason why there is a vertical asymptote. Refer to the path of the laser beam. ............................................................................................................................. ............................................................................................................................. [1]
576
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The laser source, L, now stays fixed.
At the start $AR = 6$ and $RB = 15$.
The point R then moves x metres towards B along the ground.
The dashed line shows the path of the laser beam.
(a) Show that $h = \frac{60 - 4x}{6 + x}$. [2]
(b) Sketch the graph of $h = \frac{60 - 4x}{6 + x}$ for $-6 < x \leq 15$. [3]
(c) When the point R has moved x metres towards B, the height, h, is 6.
Find the value of x. [2]
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At the start, when $AR = 6$, the height of the image is 10.
After the point $R$ moves $x$ metres, the height of the image is $h = \frac{60 - 4x}{6 + x}$.
$y$ is the change in the height of the image, so $y = 10 - h$.
(a) Show that a model for $y$ is $y = \frac{14x}{6 + x}$.
[3]
(b) (i) When the point $R$ moves one metre to the left, away from $B$, $x = -1$.
Use the model in part (a) to find the change in height of the image.
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[2]
(ii) The point $R$ moves an additional one metre to the left, away from $B$.
(a) Write down the value of $x$.
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[1]
(b) Find the additional change in height.
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[3]
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(a) Find $h$ in terms of $a$ and $b$.
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(b) The point $R$ moves $x$ metres to the right, towards $B$, along the horizontal ground. $y$ is the change in $h$.
Find a model for $y$ in terms of $a$, $b$ and $x$.
Do not simplify your answer.
................................................................. [3]
508
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556
