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GEOMETRY.

THE GRAPHICAL SOLUTION OF QUADRATIC EQUATIONS.

From the following constructions, which depend on Problem 32, a graphical solution of easy quadratic equations may be obtained.

I. To divide a straight line internally so that the rectangle contained by the segments may be equal to a given square.

Let AB be the st. line to be divided, and DE a side of the given square.

Construction. On AB draw a semicircle; and from B draw BF perp. to AB and equal to DE.

From F draw FCC' par1 to AB, cutting the Oce at C and C'.

From C, C' draw CX, C'X' perp. to AB.

Then AB is divided as required at X, and also at X'.

Application. The purpose of this construction is to find two straight lines AX, XB, having given their sum, viz. AB, and their product, viz. the square on DE.

Now to solve the equation x2 - 13x+36=0, we have to find two numbers whose sum is 13, and whose product is 36, or 62.

To do this graphically, perform the above construction, making AB equal to 13 cm., and DE equal to √36 or 6 cm. The segments AX, XB represent the roots of the equation, and their values may be obtained by measurement.

NOTE. If the last term of the equation is not a perfect square, as in x2-7x+11=0, II must be first got by the arithmetical rule, or graphically by means of Problem 32.

II. To divide a straight line externally so that the rectangle contained by the segments may be equal to a given square.

Χ' Α

B X

D

E

Let AB be the st. line to be divided externally, and DE the side of the given square.

Construction. From B draw BF perp. to AB, and equal to DE. Bisect AB at O.

With centre O, and radius OF draw a semi-circle to cut AB produced at X and X'.

Then AB is divided externally as required at X, and also at X'.

Proof.

AX. XB X'B. BX, since AX=X'B, see flow =BF2

=DE2.

Prob. 32.

Application. Here we find two lines AX, XB, having given their difference, viz. AB, and their product, viz. the square on DE.

Now to solve the equation x2-6x-16=0, we have to find two numbers whose numerical difference is 6, and whose product is 16, or 42.

To do this graphically, perform the above construction, making AB equal to 6 cm., and DE equal to √16 or 4 cm. The segments AX, XB represent the roots of the equation, and their values, as before, may be obtained by measurement.

EXERCISES.

Obtain approximately the roots of the following quadratics by means of graphical constructions; and test your results algebraically.

EXERCISES FOR SQUARED PAPER.

1. A circle passing through the points (0, 4), (0, 9) touches the x-axis at P. Calculate and measure the length of OP.

2. With centre at the point (9, 6) a circle is drawn to touch the y-axis. Find the rectangle of the segments of any chord through O. Also find the rectangle of the segments of any chord through the point (9, 12).

3. Draw a circle (shewing all lines of construction) through the points (6, 0), (24, 0), (0, 9). Find the length of the other intercept on the y-axis, and verify by measurement. Also find the length of a tangent to the circle from the origin.

4. Draw a circle through the points (10, 0), (0,5), (0, 20); and prove by Theorem 59 that it touches the x-axis.

Find (i) the coordinates of the centre, (ii) the length of the radius. 5. If a circle passes through the points (16, 0), (18, 0), (0, 12), shew by Theorem 58 that it also passes through (0, 24).

Find (i) the coordinates of the centre, (ii) the length of the tangent from the origin.

6. Plot the points A, B, C, D from the coordinates (12, 0), (-6, 0), (0, 9), (0, -8); and prove by Theorem 57 that they are concyclic. If r denotes the radius of the circle, shew that

OA2 + OB2+OC2 + OD2=4r2

7. Draw a circle (shewing all lines of construction) to touch the y-axis at the point (0, 9), and to cut the x-axis at (3, 0).

Prove that the circle must cut the x-axis again at the point (27, 0); and find its radius. Verify your results by measurement.

8. Shew that two circles of radius 13 may be drawn through the point (0, 8) to touch the x-axis; and by means of Theorem 58 find the length of their common chord.

9. Given a circle of radius 15, the centre being at the origin, shew how to draw a second circle of the same radius touching the given circle and also touching the x-axis.

How many circles can be so drawn? Measure the coordinates of the centre of that in the first quadrant.

10. A, B, C, D are four points on the x-axis at distances 6, 9, 15, 25 from the origin O. Draw two intersecting circles, one through A, B, and the other through C, D, and hence determine a point P in the x-axis such that

PA. PB PC. PD.

Calculate and measure OP.

If the distances of A, B, C, D from O are a, b, c, d respectively, prove that

OP=(ab-cd)/(a+b-c-d).

APPENDIX.

This

The solid whose shape you are probably most familiar with is that represented by a brick or slab of hewn stone. solid is called a rectangular block or cuboid.

its form more closely.

Let us examine

How many faces has it? How many edges? How many corners, or vertices?

The faces are quadrilaterals: of what shape?

Compare two opposite faces.

parallel?

Are they equal? Are they

We may now sum up our observations thus:

A cuboid has six faces; opposite faces being equal rectangles in parallel planes. It has twelve edges, which fall into three groups, corresponding to the length, the breadth, and the height of the block. The four edges in each group are equal and parallel, and perpendicular to the two faces which they cut.

The length, breadth, and height of a rectangular block are called its three dimensions.

Ex. 1. If two dimensions of a rectangular block are equal, say, the breadth AC and the height AD, two faces take a particular shape. Which faces? What shape?

Ex. 2. If the length, breadth, and height of a rectangular block are all equal, what shapes do the faces take?