## How to Solve the Thales Theorem?

- To prove the Thales theorem, draw a perpendicular bisector of ∠
- Let point M be midpoint point of line
*AC* - Also let ∠
*MBA*= ∠*BAM*= β and ∠*MBC*=∠*BCM*=α - Line
*AM*=*MB*=*MC*= the radius of the circle. - Δ
*AMB*and Δ*MCB*are isosceles triangles.

By triangle sum theorem,

∠*BAC *+∠*ACB *+∠*CBA* = 180°

β + β + α + α = 180°

Factor the equation.

2 β + 2 α = 180°

2 (β + α) = 180°

Divide both sides by 2.

**β + α = 90°. **

Therefore, ∠*ABC* = 90°, hence proved

Let’s work out a few example problems involving Thales theorem.

*Example 1*

Given that point O is the center of the circle shown below, find the value of x.

__Solution__

Given that line, *XY* is the diameter of the circle, then by Thales theorem,

∠*XYZ = *90°.

Sum of interior angles of a triangle = 180°

90° + 50° + x =180°

Simplify.

140° + x =180°

Subtract 140° on both sides.

x = 180° – 140°

x = 40°.

So, the value of x is 40 degrees.

*Example 2*

If point D is the centre of the circle shown below, calculate the diameter of the circle.

__Solution__

By Thales theorem, triangle *ABC *is a right triangle where ∠*ACB = *90°.

To find the diameter of the circle, apply the Pythagorean theorem.

CB^{2} + AC^{2} =AB^{2}

8^{2} + 6^{2} = AB^{2}

64 + 36 = AB^{2}

100 = AB^{2}

AB = 10

Hence, the diameter of the circle is 10 cm

*Example 3*

Find the measure of angle *PQR *in the circle shown below. Assume point *R* is the centre of the circle.

__Solution__

Triangle *RQS *and *PQR* are isosceles triangles.

∠*RQS *=∠*RSQ* =*64*°

By Thales theorem, ∠*PQS *=* 90*°

So, ∠*PQR *= *90*° – *64*°

= 26°

Hence, the measure of angle *PQR* is 26°.

*Example 4*

Which one of the following statements is true about definition of the Thales theorem?

A. The central angle is twice the measure of the inscribed angle

B. An angle inscribed in a half-circle will be a right angle.

C. The diameter of a circle is the longest chord.

D. The diameter of a circle is twice the length of the radius.

__Solution__

The correct answer is:

B. An angle inscribed in a half-circle will be a right angle.

*Example 5*

In the circle shown below, line *AB* is the diameter of the circle with centre *C*.

- Find the measure of ∠
*BCE* - ∠
*DCA* - ∠
*ACE* - ∠
*DCB*

__Solution__

Given triangle *ACE* is an isosceles triangle,

∠ *CEA =*∠ *CAE = 33*°

So, ∠ *ACE =180*° – (33° + 33°)

∠ *ACE *= 114°

But angles on a straight = 180°

Hence, ∠ *BCE = 180*° – 114°

= 66°

Triangle *ADC* is an isosceles triangle, therefore, ∠ *DAC *=20°

By triangle sum theorem, ∠*DCA* = 180° – (20° + 20°)

∠ *DCA* = 140°

∠ *DCB = 180*° – 140°

= 40°

*Example 6*

What is the measure of ∠*ABC*?

__Solution__

The Thales theorem states that *BAC *= 90°

And by triangle sum theorem,

∠*ABC *+ 40° + 90° = 180°

∠*ABC = *180° – 130°

= 50°

*Example 7*

Find the length of *AB* in the circle shown below.

__Solution__

Triangle ABC is a right triangle.

Apply the Pythagorean theorem to find length *AB*.

*AB ^{2}* + 12

*= 18*

^{2}^{2}

*AB ^{2}* + 144 = 324

*AB ^{2}* = 324 – 144

*AB ^{2}* = 180

*AB* = 13.4

Therefore, the length of *AB* is 13.4 cm.