A flat machine part consists of two circular ends attached to a plate, as shown (diagram not to scale). The sides of the plate, HK and PQ, are tangential to each circle. The larger circle has centre A and radius 4r cm. The smaller circle has centre B and radius r cm. The length of [HK] is 8 cm and |AB| = 20/73 cm. (a) Find r, the radius of the smaller circle. (Hint: Draw BT || KH, T ∈ AH.) (b) Find the area of the quadrilateral ABKH. (c) (i) Find ∠HAP, in degrees, correct to one decimal place. (ii) Find the area of the machine part, correct to the nearest cm².

Leaving Cert Mathematics Geometry: A flat machine part consists of

A flat machine part consists of two circular ends attached to a plate, as shown (diagram not to scale) - Leaving Cert Mathematics - Question 7 - 2015

Question 7

A flat machine part consists of two circular ends attached to a plate, as shown (diagram not to scale). The sides of the plate, HK and PQ, are tangential to each ci... show full transcript

Worked Solution & Example Answer:A flat machine part consists of two circular ends attached to a plate, as shown (diagram not to scale) - Leaving Cert Mathematics - Question 7 - 2015

Step 1

Find r, the radius of the smaller circle.

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Answer

To find r, we can use the Pythagorean theorem. Given that:

The distance from A to T is 3r
The distance from B to K is r
The tangent segments HK and AT satisfy the equation:

$AT^2 + BT^2 = |AB|^2$

Substituting the known values: $(3r)^2 + (8)^2 = \left(\frac{20}{73}\right)^2$

Expanding this: $9r^2 + 64 = \frac{400}{5329} \implies 9r^2 = 29200 \implies r^2 = 3200$

Therefore, taking the square root: $r = 20 \text{ cm}$

Step 2

Find the area of the quadrilateral ABKH.

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Answer

The area of quadrilateral ABKH can be calculated using:

$|ABKH| = |BKHT| + |ΔABT|$

With the dimensions provided: $|ABKH| = 20 \times 160 + \frac{1}{2}(60)(160)$

Calculating: $|ABKH| = 8000 \text{ cm}^2$

Step 3

Find ∠HAP, in degrees, correct to one decimal place.

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Answer

To find ∠HAP, we can use the tangent function:

$\tan |HAB| = \frac{160}{60}$

Calculating: $|HAB| = \tan^{-1}\left(\frac{160}{60}\right) = 69^{\circ}$

Thus: $|HAP| = 180 - 69 = 138^{\circ}$

Step 4

Find the area of the machine part, correct to the nearest cm².

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Answer

The total area of the machine part is found by summing the areas of the sectors and rectangles:

$ext{Area} = \frac{\pi}{180} \cdot (80)(20) + 2 \cdot 8000 + \pi(20)(9)$

Calculating:

Area of sector HKP = 12348.55 cm²
Area of rectangle ABKH = 8000 cm²
Area of sector KBQ = 2883.43 cm²

Total area rounded gives approximately: $28834 \text{ cm}^2$

A flat machine part consists of two circular ends attached to a plate, as shown (diagram not to scale) - Leaving Cert Mathematics - Question 7 - 2015

Worked Solution & Example Answer:A flat machine part consists of two circular ends attached to a plate, as shown (diagram not to scale) - Leaving Cert Mathematics - Question 7 - 2015

Find r, the radius of the smaller circle.

Find the area of the quadrilateral ABKH.

Find ∠HAP, in degrees, correct to one decimal place.

Find the area of the machine part, correct to the nearest cm².

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