9. A secant is drawn from the vertex of a square ABCD, cuting the side BC in E and the side DC produced in F so that the segment EF is b inches long. How long is the segment AE if the side of the square is a inches?

10. If one leg of a right-angled triangle is 5 inches longer than the other and both legs are produced 3 inches, a new triangle is formed whose hypotenuse is 4 inches longer than the hypotenuse of the original triangle. How long are the legs of the original triangle? 11. How long is a side of an equilateral triangle whose altitude is n inches in length?

12. How long is the longest diagonal of a rhombus whose base is 100 inches and whose shortest diagonal is 4 inches?

13. The diagonal of a square is a inches longer than a side. What is the length of the side of the square?

14. The altitude of an isosceles triangle is 3 inches longer than the base. How long are both if the equal sides are 19 inches long?

15. If an equilateral triangle is inscribed in a square of which a side is a, how long is a side of the triangle?

16. An isosceles triangle is inscribed in a square of which a side is a; one of the equal sides of the triangle is binches longer than its base. How long are the equal segments cut off from the corner of the square by the base of the triangle?

17. A circle is inscribed in a circular quadrant, tangent to the arc and the two perpendicular radii. How long is the radius of the original circle, if the radius of a new circle is a inches shorter than that of the given circle?

18. A circular quadrant is circumscribed about a circle of radius a so that the radii and the bounding arc of the quadrant are all tangent to the given circle. How long is the radius of the quadrant?

19. In a right-angled triangle, the median drawn to one of the legs is a inches longer than this leg. How long is this leg if the other is b inches?

20. One of the legs of a right-angled triangle is a inches and the median drawn to the leg is 6 inches shorter than the hypotenuse. How long is the hypotenuse?

21. If the altitude of an isosceles triangle is prolonged through the vertex a distance equal to its own length and the extremity of the extension joined with an extremity of the base, the joining line will be a inches longer than the side of the isosceles triangle. How long is the altitude, if the base is b?

420. Problems Concerning the Area of Plane Figures.

PROBLEM I.-Within a given rectangle whose sides are a and b, a second rectangle is constructed so that its sides are everywhere equally distant from the sides of the given rectangle, and that it has one-half the area of the given rectangle. How long is the perimeter of the second rectangle?

Since only the lower sign of the radical can be used, one obtains for the perimeter

2 (ab 4x) = 21 a2 + b2.

PROBLEM II-A right-angled triangle whose legs are a and b is divided into three equal parts by lines perpendicular to the hypotenuse. Into what parts is the hypotenuse divided?

If the perpendiculars ED and GF divide the triangle into three equal parts, then the area of the triangle ADE is ab. Since this triangle is similar to the triangle ACH, one has, if he puts x = AD,

6

PROBLEM III. - A line is drawn from the corner B of a square ABCD to a point E on the non-adjacent side CD so that its length is a inches longer than a side of the square. The area of the triangle cut off has to the area of the trapezoidal figure remaining the ratio b: c. How long is the side of the square?

4. It is desired to frame a rectangular mirror, whose sides are a and b, so that the area of the frame shall equal that of the mirror. What must the width of the frame be if the mirror framed is similar in shape to the unframed mirror?

5. Of two sides of a triangle, whose area is 468 square inches, one is 1 inch longer than the other. If the third side is 25 inches long, how long are the first two sides?

Explanation.-Let the three sides of the triangle be a, b, c, and

Then the area of the triangle will be represented by

V ́s (s — a) (s — b) (s — c).

Let x and x + 1 be two of the sides of the triangle; then

6. Two sides of a triangle are in the ratio 3:5; the is 2 cm. long; the area of the triangle is 150 sq. mm. are the other two sides?

third side How long

Explanation.-Let the two sides a and b be in the ratio 3 : 5,

and put

a =

= 3x, b

=

5x, c = 20 mm., etc.

7. If the area of a triangle is d2 and two of its sides are a and b, find the value of the third side.

8. A triangle, two of whose sides are a and b and whose altitude on the third side is h, is divided by lines parallel to the altitude into three equal parts. How long are the three segments of the third side? (Compare solution of Problem II, page 418.)

DG = (V a2 -

h2

va2

· T2 + v b2 — k ) ( 1 − ¦1/3 V væ− h + V b2 — h2).

9. From a point without a circle, two tangents are drawn to the circle; the two tangents and the radii drawn to the points of contact form a quadrilateral whose area is a square inches. How long is the radius of the circle if the chord which connects the points of contact is 2 inches long?

10. A frame is made for a mirror. The area of the mirror is a2 square inches, the frame on all sides is 6 inches wide, and the perimeter of the frame is four times as long as the perimeter of the mirror. How long are the sides of the unframed mirror?

11. Two tangents to a circle intersect in a point which is a inches from the center, and the radii drawn to the points of contact form a quadrilateral, whose area is b square inches. How long are the tangents?

12. How long is the side of a regular decagon whose area is a2 square inches?

13. Within a square whose side is a, a second square is so con. structed that its sides are at a given distance from those of the first. Within the second square a third is constructed so that its sides are at the same distance from the sides of the second square, and a fourth is similarly constructed within the third. How great is this distance if the sum of the areas of the second, third, and fourth squares is equal to that of the first?

14. A circle is inscribed in, and another is circumscribed about, a square. The ring bounded by the two circles contains a2 square inches. How long is the side of the square?

15. A square is inscribed in, and another is circumscribed about, a circle. The area of the figure between the two squares is a2 square How long is the radius of the circle?

inches.