𝐴, 𝐵, 𝐶, and 𝐷 are points on
the circumference of a circle. 𝐴𝑃𝐶 and 𝐷𝑃𝐵 are straight
lines. The following lengths are given:
𝐴𝐷 is equal to 57 over four, 𝑃𝐷 is equal to 12, and 𝐶𝑃 is equal to four. Based on this information, find the
length of 𝐵𝐶.
Using our angle properties, we can
see that angle 𝐴𝑃𝐷 is equal to angle 𝐵𝑃𝐶 as they are opposite angles. The angles 𝐷𝐴𝑃 and 𝑃𝐵𝐶 are
also equal as they have a shared arc — the arc 𝐷𝐶. This means that they are angles in
the same segment and they’re therefore equal. The angles 𝐴𝐷𝑃 and 𝑃𝐶𝐵 are
equal for the same reason. This time their shared arc is the
arc 𝐴𝐵. We can therefore say that triangle
𝐴𝑃𝐷 and triangle 𝐵𝑃𝐶 are similar as all three angles are equal.
In order to work out the length
𝐵𝐶, we firstly need to find the scale factor. As triangle 𝐵𝑃𝐶 is smaller than
triangle 𝐴𝑃𝐷, the scale factor will be a reduction. The scale factor is equal to four
twelfths. This can be simplified to
one-third. Therefore, triangle 𝐵𝑃𝐶 is a
third of the size of triangle 𝐴𝑃𝐷. In order to calculate the length of
𝐵𝐶 or 𝑥, we need to multiply 57 divided by four — fifty-seven quarters — 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 𝐵𝐶
is 19 over four or nineteen quarters. This could be rewritten as a
decimal as 4.75.