## 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°.

## Distance between projection points on the legs of right triangle (solution by Calculus)

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?

## 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.

Problem 33
A lot has the form of a right triangle, with perpendicular sides 60 and 80 feet long. Find the length and width of the largest rectangular building that can be erected, facing the hypotenuse of the triangle.

Problem 34
Solve Problem 33 if the lengths of the perpendicular sides are a, b.

## 21 - 24 Solved problems in maxima and minima

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

Problem 22
If the hypotenuse of the right triangle is given, show that the area is maximum when the triangle is isosceles.

Problem 23
Find the most economical proportions for a covered box of fixed volume whose base is a rectangle with one side three times as long as the other.

## 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.

## 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.

## 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?

## Trigonometry

### Video

The Six Trigonometric Functions

1. $\sin \theta = \dfrac{a}{c}$
2. $\cos \theta = \dfrac{b}{c}$
3. $\tan \theta = \dfrac{a}{b}$
4. $\csc \theta = \dfrac{c}{a}$
5. $\sec \theta = \dfrac{c}{b}$
6. $\cot \theta = \dfrac{b}{a}$