Question

Q14)

Points A, B and C lie on the circumference of a circle and DE is the
tangent to the circle at A. Angle DAB = 79º and angle EAC = 37º.
2
Find the sizes of angles x and y.
Not to scale
Give a reason for your answers.
B
7
[Total 3
790
1370
E
A

Answer 1

x = 37

y = 79

Explanation:

Remark

We are going to go through an extensive discussion of the value of x. The value for y is found exactly the same way.

Top diagram.

The top diagram establishes the fact that if any two points are on the major arc of the circle whose center is 0, then the vertex of the of the two angles so formed are equal. The end points of these two angles must be established by the endpoints of the chord A and B.

Put in much simpler language.

<ACB = <AC'B

The end points of the angles are A and B of the Chord AB.

The Vertex of the angles are both on the major arc of the circle. These two are C and C'

Bottom Diagram

The idea here is to find the value of C. If we do that, then we know what x is in the diagram you provided.

Draw diameter COA Construction

< CAB = 90 Any vertex (C) on circumference with diameter endpoints (A & C) = 90

<CAT_1 = 90 A diameter always meets a tangent at 90o

<BAT_1 = 37 Given

<CAB = 90-37 <CAB and <BAT_1 are complementary

<ACB = 37 A right triangle's acute angles are complementary.

<ACB = x = 37o See top diagram