# Video: Pack 1 β’ Paper 2 β’ Question 12

Pack 1 β’ Paper 2 β’ Question 12

02:28

### Video Transcript

π΄, π΅, πΆ, and π· are points on the circumference of a circle. π΄ππΆ and π·ππ΅ are straight lines. The following lengths are given: π΄π· is equal to 57 over four, ππ· is equal to 12, and πΆπ is equal to four. Based on this information, find the length of π΅πΆ.

Using our angle properties, we can see that angle π΄ππ· is equal to angle π΅ππΆ as they are opposite angles. The angles π·π΄π and ππ΅πΆ are also equal as they have a shared arc β the arc π·πΆ. This means that they are angles in the same segment and theyβre therefore equal. The angles π΄π·π and ππΆπ΅ are equal for the same reason. This time their shared arc is the arc π΄π΅. We can therefore say that triangle π΄ππ· and triangle π΅ππΆ are similar as all three angles are equal.

In order to work out the length π΅πΆ, we firstly need to find the scale factor. As triangle π΅ππΆ is smaller than triangle π΄ππ·, the scale factor will be a reduction. The scale factor is equal to four twelfths. This can be simplified to one-third. Therefore, triangle π΅ππΆ is a third of the size of triangle π΄ππ·. In order to calculate the length of π΅πΆ or π₯, we need to multiply 57 divided by four β fifty-seven quarters β by one-third. Simplifying this calculation gives us 19 quarters multiplied by one. 19 multiplied by one is 19 and four multiplied by one is equal to four.

This means that the length of π΅πΆ is 19 over four or nineteen quarters. This could be rewritten as a decimal as 4.75.