May 2012

714 Inverted T-section | Centroid of Composite Figure

Problem 714
The dimensions of the T-section of a cast-iron beam are shown in Fig. P-714. How far is the centroid of the area above the base?
 

Inverted T-section for centroid problem

 

715 Semicircle and Triangle | Centroid of Composite Figure

Problem 715
Determine the coordinates of the centroid of the area shown in Fig. P-715 with respect to the given axes.
 

Semicircle surmounted on top of a right triangle

 

716 Semicircular Arc and Lines | Centroid of Composite Figure

Problem 716
A slender homogeneous wire of uniform cross section is bent into the shape shown in Fig. P-716. Determine the coordinates of the centroid.
 

Vertical line, semicircular arc, and 30 deg line

 

717 Symmetrical Arcs and a Line | Centroid of Composite Line

Problem 717
Locate the centroid of the bent wire shown in Fig. P-717. The wire is homogeneous and of uniform cross-section.
 

A line and two arcs in vertical symmety

 

719 Closed Straight Lines | Centroid of Composite Lines

Problem 719
Determine the centroid of the lines that form the boundary of the shaded area in Fig. P-718.
 

Trapezoidal area with isosceles triangle subtracted from the bottom

 

08 Circles with diameters equal to corresponding sides of the triangle

Problem
From the figure shown below, O1, O2, and O3 are centers of circles located at the midpoints of the sides of the triangle ABC. The sides of ABC are diameters of the respective circles. Prove that
 

$A_1 + A_2 + A_3 = A_4$

 

where A1, A2, A3, and A4 are areas in shaded regions.
 

Circles with centers at midpoints of sides of a right triangle

 

09 Areas outside the overlapping circles indicated as shaded regions

Problem
From the figure shown, AB = diameter of circle O1 = 30 cm, BC = diameter of circle O2 = 40 cm, and AC = diameter of circle O3 = 50 cm. Find the shaded areas A1, A2, A3, and A4 and check that A1 + A2 + A3 = A4 as stated in the previous problem.
 

Circles with centers at midpoints of sides of a right triangle

 

10 Area common to three squares inside the regular hexagon

Problem
Three squares are drawn so that each will contain a side of regular hexagon as shown in the given figure. If the hexagon has a perimeter of 60 in., compute the area of the region common to the three squares. The required area is the shaded region in the figure.
 

Equilateral triangle bounded by three squares

 

01 Area enclosed by line rays inside a square

Problem
The figure shown below is a square of side 4 inches. Line rays are drawn from each corner of the square to the midpoints of the opposite sides. Find the area of the shaded region.
 

Four-pointed star inside the square

 

09 Dimensions of smaller equilateral triangle inside the circle

Problem
From the figure shown, ABC and DEF are equilateral triangles. Point E is the midpoint of AC and points D and F are on the circle circumscribing ABC. If AB is 12 cm, find DE.
 

Two equilateral triangles inside a circle

 

01 Minimum distance between projection points on the legs of right triangle

Problem
From the right triangle ABC shown below, AB = 40 cm and BC = 30 cm. Points E and F are projections of point D from hypotenuse AC to the perpendicular legs AB and BC, respectively. How far is D from AB so that length EF is minimal?
 

030-projections-of-d.gif