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Figure 1 shows a pictorial view of a bracket. Draw the following views in first angle projection: a) Front elevation from arrow direction b) An end elevation c) A plan. Include six major dimensions. Show hidden details

Mathematics

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Here are the construction steps for question 33:

33. a) Construct a regular pentagon within a circle given the diagonal of 82 mm.

First, we need to determine the radius of the circumscribing circle. For a regular pentagon, the relationship between the diagonal (dd) and the circumradius (RR) is given by the formula:

d=2Rsin⁡(72∘)d = 2R \sin(72^\circ)

Given the diagonal d=82 mmd = 82 \text{ mm}, we can calculate the radius RR:

R=d2sin⁡(72∘)R = \frac{d}{2 \sin(72^\circ)} R=822×0.9510565R = \frac{82}{2 \times 0.9510565} R=821.902113R = \frac{82}{1.902113} R≈43.11 mmR \approx 43.11 \text{ mm}

Now, follow these steps to construct the pentagon:

  1. Draw a circle with center O and radius R=43.11 mmR = 43.11 \text{ mm}.

  2. Draw a horizontal diameter AB through O.

    Construct a vertical radius OC perpendicular to AB.

  3. Bisect the radius OA at point M.

  4. With M as the center and radius MC, draw an arc that intersects the diameter AB at point N.

  5. The length CN is the side length of the regular pentagon.

    With C as the center and radius CN, draw an arc that intersects the circle at points D and E.

  6. With D as the center and radius CN, draw an arc that intersects the circle at point F.

  7. With E as the center and radius CN, draw an arc that intersects the circle at point G.

  8. Connect the points C, D, F, G, E in order to form the regular pentagon.

    33. b) Construct a triangle with base angles 60° and 45° and an altitude of 76 mm.

    Let the triangle be ABC, with base BC and apex A. Let D be the foot of the altitude from A to BC, so AD is the altitude.
    Given altitude AD=76 mmAD = 76 \text{ mm}, base angle ∠B=60∘\angle B = 60^\circ, and base angle ∠C=45∘\angle C = 45^\circ.
    In right-angled triangle ABD, ∠BAD=90∘−∠B=90∘−60∘=30∘\angle BAD = 90^\circ - \angle B = 90^\circ - 60^\circ = 30^\circ.
    In right-angled triangle ACD, ∠CAD=90∘−∠C=90∘−45∘=45∘\angle CAD = 90^\circ - \angle C = 90^\circ - 45^\circ = 45^\circ.

    Follow these steps to construct the triangle:

  9. Draw a horizontal line, L1L_1.

    This line will contain the base BC of the triangle.

  10. Choose an arbitrary point D on L1L_1.

    Construct a perpendicular line, L2L_2, to L1L_1 at point D.

  11. Measure 76 mm along L2L_2 from D to locate point A.

    Point A will be the apex of the triangle.

  12. From point A, draw a line segment such that it makes an angle of 30∘30^\circ with the line segment AD (on one side of AD).

    This line segment will intersect L1L_1 at point B.

  13. From point A, draw another line segment such that it makes an angle of 45∘45^\circ with the line segment AD (on the other side of AD).

    This line segment will intersect L1L_1 at point C.

  14. Connect points A, B, and C to form the required triangle ABC.

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