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Ex. 756. Find the area of a parallelogram whose base is 15 inches and whose altitude is 2 feet.

Ex. 757. Two sides of a parallelogram are 15 and 20 respectively, and include an angle of 30°. Find the area.

Ex. 758. The sides of a parallelogram are 5 and 8 respectively, and the projection of 5 upon 8 is 3. Find the area.

Ex. 759. To divide a parallelogram into three equivalent parts.

PROPOSITION V. THEOREM

346. The area of a triangle is equal to one-half the product of its base and altitude.

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Hyp. In A ABC, the base is b, and the altitude h.

To prove

area of ABC = { b x h. Proof. Construct the parallelogram ABDC.

The diagonal of a parallelogram divides it into two equal triangles.

.. A ABC = 4ABDC. But

area ABDC=b x h.
. . area ABC = 1 b x h.

Q.E.D.

347. Cor. 1. Triangles having equal bases and equal altitudes are equivalent.

348. Cor. 2. Any two triangles are to each other as the products of their bases and altitudes.

349. Cor. 3. Triangles having equal bases are to each other as their altitudes.

350. Cor. 4. Triangles having equal altitudes are to each other as their bases.

351. DEF. To transform a figure means to find another figure equivalent to the given one.

Ex. 760. If A denotes the area and 2 s the perimeter of A abc, then A=V8(8 – a)(s b)(8 c).

Ex. 761. To transform a given triangle into another one, having the same base.

Ex. 762. What is the locus of the vertices of all the equivalent triangles constructed on the same base ?

Ex. 763. To transform a given triangle into a right triangle.

Ex. 764. To transform A ABC into an isosceles triangle, having its base equal to AB.

Ex. 765. To transform A ABC into an isosceles triangle, having an arm equal to AB.

Ex. 766. In Exs. 761-765, how many parts of the given triangle are not altered by the transformation ?

Ex. 767. To transform a given triangle into another one, having given one side.

Ex. 768. To transform a given triangle into a right triangle, having the hypotenuse equal to one side of the given triangle.

Ex. 769. To transform a given triangle into another one, having its vertical angle equal to a given angle.

352. REMARK. - Cor. 1 of Prop. V is the principal means for transformation of figures. Frequently two transformations are necessary.

Ex. 770. To transform A ABC into another triangle, having two of its sides respectively equal to two given lines m and n.

Ex. 771. To transform A ABC into another triangle, having a side equal to a given line m, and one adjacent angle equal to a given angle A.

Ex. 772. To transform a given triangle into a right triangle, having a given arm.

Ex. 773. To transform A ABC into a right triangle, having the hypotenuse equal to a given line m.

Ex. 774. To transform A ABC into an isosceles triangle, having the base equal to a given line m.

Ex. 775. To transform A ABC into an isosceles triangle, having an arm equal to a given line m.

Ex. 776. To transform A ABC into a triangle having one side equal to a given line m, and the opposite angle equal to a given angle A.

Ex. 777. To transform a parallelogram into a rectangle.
Ex. 778. To transform a parallelogram into a rhombus.

Ex. 779. To transform a parallelogram into another one having a given side.

Ex. 780. To transform a parallelogram into another one containing a given angle.

Ex. 781. To transform a parallelogram into a rectangle, having one side equal to a given line.

Ex. 782. To divide a triangle ABC into three equivalent parts by two lines passing through A.

Ex. 783. To divide a given parallelogram into two equivalent parts by a line parallel to the bases.

Ex. 784. To divide a given parallelogram into two equivalent parts by a line perpendicular to the bases.

Ex. 785. To divide a given parallelogram into two equivalent parts by a line parallel to a given line.

Ex. 786. To transform a given quadrilateral ABCD into another one, having three of its vertices in A, B, and C, respectively.

Ex. 787. To transform a given quadrilateral into a triangle.

Ex. 788. To transform A ABC into a triangle containing the angle A, and having one vertex in D, if D is a given point in AC.

Hint. — Consider ABCD a quadrilateral, and apply Ex. 787.

Ex. 789. The diagonals divide a parallelogram into four equivalent triangles.

Ex. 790. Two triangles are equivalent if two sides of the one are respectively equal to two sides of the other, and the included angles are supplementary.

Ex. 791. The lines joining the midpoint of a diagonal of a quadrilateral with the opposite vertices divide the figure into two equivalent parts.

Ex. 792. To divide a quadrilateral into three equivalent parts.

Ex. 793. The area of a triangle is 600 square inches, and the altitude is 20 inches. Find the base.

Ex. 794. Two sides of a triangle are 5 and 8, respectively, and include an angle of 30°. Find the area.

Ex. 795. Find the area of a triangle whose sides are respectively (a) 13, 14, 15, (6) 9, 10, 17, (c) 11, 25, 30.

Ex. 796. To construct a triangle three times as large as a given triangle.

PROPOSITION VI. PROBLEM 353. To transform a polygon into a triangle.

B

A

E

Given. Polygon ABCDE.
Required. To construct a triangle equivalent to ABCDE.
Construction. Draw CE
Through D, draw DF || CE, meeting AE produced in F.
Then polygon ABCF is = to ABCDE, and has one less side.
Continue the process until a triangle is obtained.
[The proof is left to the student.]

Ex. 797. To transform a parallelogram into a right triangle.
Ex. 798. To transform a trapezoid into an isosceles triangle.

PROPOSITION VII. THEOREM 354. The area of a trapezoid is equal to one-half the product of its altitude and the sum of its bases.

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To prove

Hyp. Trapezoid ABCD has the bases b and c respectively, and the altitude h.

area ABCD = { h (b + c). Proof. Draw AC. Area ABC = ih x b.

(346) Area DCA = 1h x c.

(346) .. area ABCD= 1 h(b + c).

Q.E.D.

Ex. 799. A line joining the midpoints of the bases of a trapezoid divides the trapezoid into two equivalent parts.

Ex. 800. The bases of a trapezoid are 12 and 8 respectively, and the altitude is 5. Find the area.

Ex. 801. In the diagram for Prop. VII, if b = 6, c=4, BC = 10, and 2B = 30°. Find the area.

Ex. 802. The area of a trapezoid is 200, the bases are 15 and 25 respectively. Find the altitude.

Ex. 803. The area of a trapezoid is 30, the altitude is 5, and one base is 8. Find the other base.

Ex. 804. In the diagram for Prop. VII if E, the midpoint of DA, be joined to C and B, A CBE is one-half of ABCD.

Ex. 805. If the three altitudes of a triangle are equal, the triangle is equilateral.

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