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Angles A, B and C to the horizontal = 3 tan -1   = 56.3 (accurate to one 2 decimal places)

y-x=4 Mathematical notes

That the angles A, B and C are all equal is sometimes disputed by students who measure the angles with a protractor.

Comparing the angle and gradient of a line

Similar triangles

A 3 D

8

2

B

x

E 7

C

A 7

If angles ADE and ABC are equal, as shown above, then triangles ABC and ADE are similar. (Angle DAE is common to both triangles; angles AED and ACB are right. These two facts are sufficient to show similarity.) In similar triangles the sides are proportional.

8 = 1.14 (accurate 7 to two decimal places)

Gradient of line =

Angle of line to horizontal (A) = 8 tan -1   = 48.8 (accurate to one 7 decimal places)

To find x: AE:ED = AC:CB 3:2 = 7:x 2 x = 7  3

C x=

3 B 2

14 2 =4 3 3

An alternative approach, which has proved clearer to GCSE candidates, is:

A

1

www.inquirymaths.org © 2005 Andrew Blair

2

x3

3 x

2

7 3

x

2

x

x3

7

7 3

2 7 Either x = 7   or x = 2   3 3

Using Pythagoras’ Theorem to find the length of the hypotenuse of triangles ABC and ADE: For triangle ADE, Length AD = 32  22 =

13

Similarly for triangle ABC, Length AB =

 14  72    3

=

49 

=

637 = 9

2

196 9

637 3

To show the ratio AD:AB = 3:7

=

13 :

637 3

117 : 9

637 3

= 117 : 3

637 3

= 117 : 637 Writing the numbers in prime factor form: = 3x3x13 : 7 x7 x13 = 3 13 : 7 13 =3:7

2