right triangle

04 Largest Right Triangle of Given Hypotenuse

Problem
Find the area of the largest right triangle whose hypotenuse is fixed at c.
 

03-largest-right-triangle-given-hypotenuse.gif           03-largest-right-triangle-given-hypotenuse-theta.gif

 

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

 

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

 

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

 

40 - Base angle of a growing right triangle

Problem 40
The base of a right triangle grows 2 ft/sec, the altitude grows 4 ft/sec. If the base and altitude are originally 10 ft and 6 ft, respectively, find the time rate of change of the base angle, when the angle is 45°.
 

32 - 34 Maxima and minima problems of a rectangle inscribed in a triangle

Problem 32
Find the dimension of the largest rectangular building that can be placed on a right-triangular lot, facing one of the perpendicular sides.

21 - 24 Solved problems in maxima and minima

Problem 21
Find the rectangle of maximum perimeter inscribed in a given circle.
 

Functions of a Right Triangle

From the right triangle shown below,
 

Right triangle with sides a, b, and c and angle theta

 

the trigonometric functions of angle θ are defined as follows:

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