Ex. 1697. On bases of 5 in. and 3 in. describe two similar triangles; calculate their areas, and find the ratio of their areas. Is it 5: 3?

▲ ABD = ||ogram ABCD, and ▲ A'B'D' = |ogram A'B'C'D'.

The parallelograms ABCD, A'B'C'D' are divided up into congruent parallelograms; the squares are divided up into congruent squares.

Ex. 1698. The ratio of corresponding altitudes of similar triangles

is equal to the ratio of corresponding sides.

THEOREM 6.

The ratio of the areas of similar triangles is equal to the ratio of the squares on corresponding sides.

Ex. 1699.

What is the ratio of the areas of two similar triangles on bases of 3 in. and 4 in.?

Ex. 1700. The area of a triangle with a base of 12 cm. is 60 sq. cm. ; find the area of a similar triangle with a base of 9 cm.

What is the area of a similar triangle on a base of 9 in.?

Ex. 1701. The areas of two similar triangles are 100 sq. cm. and 64 sq. cm.; the base of the greater is 7 cm.; find the base of the smaller.

Ex. 1702. The areas of two similar triangles are 97.5 sq. cm. and 75.3 sq. cm.; the base of the first is 17.2 cm.; find the base of the second.

Ex. 1703. The sides of a triangle ABC are 7·2 in., 3·5 in., 5.7 in.; the sides of a triangle DEF are 7·2 cm., 3.5 cm., 5.7 cm.; find the ratio of the area of the first triangle to that of the second.

Ex. 1704. Find the ratio of the bases of two similar triangles one of which has double the area of the other.

Draw two such triangles taking 1 in. as the base of the smaller.

Ex. 1705. Describe equilateral triangles on the side and diagonal of a square; find the ratio of their areas.

Ex. 1706. Draw a straight line parallel to the base of a triangle to bisect the triangle.

Ex. 1707. Describe equilateral triangles on the sides of a right-angled triangle whose sides are 1.5 in., 2 in., 2.5 in. What connection is there between the areas of the three equilateral triangles?

Ex. 1708. Prove that, if similar triangles are described on the three sides of a right-angled triangle, the area of the triangle described on the hypotenuse is equal to the sum of the other two triangles.

Ex. 1709. ABC, DEF are two triangles in which B=LE; prove that ▲ ABC: A DEF AB. BC: DE. EF.

[Draw AX 1 to BC, and DY 1 to EF.]

Ex. 1710. What is the ratio of the areas of two circles whose radii are R, r? 3 in., 2 in. ?

Ex. 1711. Draw two similar quadrilaterals ABCD, PQRS; calculate their areas (join AC, PR); find the ratio of their areas, and compare this with the ratio of corresponding sides.

RECTANGLE PROPERTIES.

What

Ex. 1712. XYZ is a triangle inscribed in a circle, XN is an altitude of XY YD the triangle and XD a diameter of the circle; prove that XN NZ rectangle property can be obtained from this by clearing of fractions?

Ex. 1713. With the same construction as in Ex. 1712, prove that

XZ. NY=XN. ZD.

[You will have to pick out two equal ratios from two equiangular triangles. If you colour XZ, NY red and XN, ZD blue you will see which are the triangles.]

Ex. 1714. ABCD is a quadrilateral inscribed in a circle; its diagonals intersect at X. Prove that (i) AX. BC=AD. BX, (ii) AX. XC=BX. XD.

Ex. 1715. ABCD is a quadrilateral inscribed in a circle; AB, DC produced intersect at Y. Prove that

(i) YA. BD =YD. CA, (ii) YA. YB YC. YD.

Ex. 1716. The rectangle contained by two sides of a triangle is equal to the rectangle contained by the diameter of the circumcircle and the altitude drawn to the base.

[Draw the diameter through the vertex at which the two sides intersect.]

Ex. 1717. The bisector of the angle A of A ABC meets the base in P and the circumcircle in Q. Prove that the rectangle contained by the sides AB, AC rect. AP. AQ.

Ex. 1718. In Ex. 1680, prove that PQ. SR=PR. TQ.

Ex. 1719. The sum of the rectangles contained by opposite sides of a cyclic quadrilateral is equal to the rectangle contained by its diagonals. (Ptolemy's theorem.)

[Use the construction of Ex. 1680.]

Ex. 1720. Draw a circle of radius 7 cm.; mark a point P 3 cm. from the centre O; through P draw five or six chords APB, CPD, .... Measure their segments and calculate the products PA. PB; PC.PD; Take the mean of your results and estimate by how much per cent. each result differs from the mean. (Make a table.)

....

Ex. 1721. Draw a circle of radius 7 cm. and mark a point P 10 cm. from the centre O; through P draw a number of chords of the circle, and proceed as in Ex. 1720.

[Remember that if P is in the chord AB produced, PA, PB are still regarded as the segments into which P divides AB; you must calculate PA. PB, not PA. AB.]

Ex. 1722. What will be the position of the chord in Ex. 1721 when the two segments are equal?