Question

In the given diagram(see image) , TAX and TBY are tangents to the circle and C is a point on the major arc.

Question :
If <CAX = 65° , <CBY = 76° , Calculate <ATB.​

Answer 1

∠ATB = 102°

Explanation:

Since ∠CBY is less than 90°, CB is not a diameter.

I did this in my head so I drew a sketch to show that around about way I did it. sketch attached:

I drew a third line from X to Y that is tangent to the circle at C.

Because each pair of tangent lines to a common point are equal, we have made three isosceles triangles.

Now we know 2 of the three angles to ΔXYT, 180 - 50 - 28 = 102° for ∠ATB

Answer 2

• 102°

Explanation:

Given

• ∠CAX = 65° , ∠CBY = 76°

To find

• ∠ATB

Solution

Refer to attached picture

• ∠CAX= ∠CBA as same arc intercepted
• ∠CBY = ∠CAB as same arc intercepted

TAX is a straight angle and:

• ∠TAB = 180° - (∠CAX + ∠CAB) = 180° - (65° + 76°) = 39°

∠TBA = ∠TAB as two tangents intersect at same distance, their opposite angles are therefore same. ΔATB is an isosceles triangle.

• ∠ATB = 180° - 2*39° = 102°