437. If through the middle point of the median of a trapezoid a line be drawn, cutting the bases, the two parts are equivalent.

438. In every trapezoid the triangle which has for its base one leg, and for its vertex the middle point of the other leg, is equivalent to one-half the trapezoid.

439. If any point within a parallelogram, selected at random, be joined to the four vertices, the sum of the areas of either pair of opposite triangles is equivalent to one-half the parallelogram.

440. The area of a trapezoid is equal to the product of one of its legs and the distance of this leg from the middle point of the other.

441. The triangle whose vertices are the middle points of the sides of a given triangle is equivalent to one-fourth the latter.

442. The parallelogram formed in 101 is equivalent to onehalf the quadrilateral.

443. If two parallelograms have two contiguous sides respectively equal, and their included angles supplementary, the parallelograms are equivalent.

444. The lines joining the middle point of either diagonal of a quadrilateral to the opposite vertices, divide the quadrilateral into two equivalent parts.

445. The line which joins the middle points of the bases of a trapezoid divides the trapezoid into two equivalent parts.

PROBLEMS OF COMPUTATION.

446. Compute the area of a right isosceles triangle if the hypothenuse is 100 rods.

447. Compute the area of a rhombus if the sum of its diagonals is 12 inches and their ratio is 3:5.

448. Compute the area of a right triangle whose hypothenuse is 13 feet and one of whose legs is 5 feet.

449. Compute the area of an equilateral triangle, one of whose sides is 40 feet.

450. The area of a trapezoid is 34 acres; the sum of the two parallel sides is 242 yards. Find the perpendicular distance between them.

451. The diagonals of a rhombus are 24 feet and 40 feet respectively. Compute its area.

452. The diagonals of a rhombus are 88 feet and 234 feet respectively. Compute its area, and find length of one of its sides.

453. The area of a rhombus is 354,144 square feet, and one diagonal is 672 feet. Compute the other diagonal and one side.

454. The sides of a right triangle are in the ratio of 3, 4, and 5, and the altitude upon the hypothenuse is 20 yards. Compute the area.

455. Compute the area of a quadrilateral circumscribed about a circle whose radius is 25 feet and the perimeter of the quadrilateral 400 feet.

456. Compute the area of a hexagon having the same length of perimeter and circumscribed about the same circle.

457. The base of a triangle is 75 rods and its altitude 60 rods. Find the perimeter of an equivalent rhombus if its altitude is 45 rods.

458. Upon the diagonal of a rectangle 40 yards by 25 yards an equivalent triangle is constructed. Compute its altitude.

459. Compute the side of a square equivalent to a trapezoid whose bases are 56 feet and 44 feet respectively, and each of whose legs is 10 feet.

460. Find what part of the entire area of a parallelogram will be the area of the triangle formed by drawing a line from one vertex to the middle point of one of the opposite sides.

461. In two similar polygons two homologous sides are 15 feet and 25 respectively. The area of the first polygon is 450 square feet. Compute the area of the other polygon.

462. The base of a triangle is 32 feet and its altitude 20 feet. Compute the area of the triangle formed by drawing a line parallel to the base at a distance of 15 feet from the base.

463. The sides of two equilateral triangles are 20 and 30 yards respectively. Compute the side of an equilateral triangle equivalent to their sum.

464. If the side of one equilateral triangle is equivalent to the altitude of another, what is the ratio of their areas?

465. The radius of a circle is 15 feet, and through a point 9 inches from the centre any chord is drawn. What is the product of the two segments of this chord?

466. A square field contains 5 acres. fence that incloses it.

Find the length of

467. A square field 210 yards long has a path round the inside of its perimeter which occupies just one-seventh of the whole field. Compute the width of the path.

468. A street 14 miles long contains 5 acres. How wide is

the street?

469. The perimeter of a rectangle is 72 feet, and its length is twice its breadth. What is its area?

470. A chain 80 feet long incloses a rectangle 15 feet wide. How much more area would it inclose if the figure were a square?

471. The perimeter of a square, and also of a rectangle whose length is four times its breadth, is 400 yards. Compute the difference in their areas.

472. A rectangle whose length is 25m is equivalent to a square whose side is 15m. Which has the greater perimeter, and how much?

473. The perimeters of two rectangular lots are 102 yards and 108 yards respectively. The first lot is & as wide as it is long, and the second lot is twice as long as it is wide. Compute the difference in the value of the two lots at $1 per square foot.

474. A rhombus and a rectangle have equal bases and equal Compute their perimeters if one side of the rhombus is 15 feet and the altitude of the rectangle is 12 feet.

areas.

475. The altitudes of two triangles are equal, and their bases are 20 feet and 30 feet respectively. Compute the base of a triangle equivalent to their sum and having an altitude as great.

REGULAR POLYGONS AND CIRCLES.

See 328.

476. An equilateral polygon inscribed in a circle is regular. Sug. See Theorem 193.

477. An equiangular polygon circumscribed about a circle is regular.

478. If a polygon be regular, a circle can be circumscribed about it; i.e. one circumference can be constructed which shall pass through all its vertices.

Post. Let ABCDH be a regular polygon of n sides.

We are to prove that a circum- A ference can be constructed which will pass through all its vertices. Dem. A circumference may be passed through any three vertices, as A, B, and C. (See Theorem 218.)

Ο

From the centre, K, of this circumference draw lines to all the vertices.

H

What relation exists between the two triangles AKB and BKC? Why?

What relation, then, exists between the two angles ABK and BCK? Why?

What relation must consequently exist between the two angles KBC and KCD? Why?

How, then, do the two triangles KBC and KCD compare? Why?

What must therefore be the relation between KD and KC? Why?

What must be true, then, of the circumference passing through the vertices A, B, and C, as regards the vertex D?

In a similar manner, fix the position of the circumference with regard to each of the other vertices.

479. If a polygon be regular, a circle may be inscribed in it; i.e. a circle may be constructed which shall have the sides of the polygon as tangents.

Sug. First circumscribe a circle by Theorem 478, then consult 214.