Video: Pack 1 β€’ Paper 2 β€’ Question 12

Pack 1 β€’ Paper 2 β€’ Question 12

02:28

Video Transcript

𝐴, 𝐡, 𝐢, and 𝐷 are points on the circumference of a circle. 𝐴𝑃𝐢 and 𝐷𝑃𝐡 are straight lines. The following lengths are given: 𝐴𝐷 is equal to 57 over four, 𝑃𝐷 is equal to 12, and 𝐢𝑃 is equal to four. Based on this information, find the length of 𝐡𝐢.

Using our angle properties, we can see that angle 𝐴𝑃𝐷 is equal to angle 𝐡𝑃𝐢 as they are opposite angles. The angles 𝐷𝐴𝑃 and 𝑃𝐡𝐢 are also equal as they have a shared arc β€” the arc 𝐷𝐢. This means that they are angles in the same segment and they’re therefore equal. The angles 𝐴𝐷𝑃 and 𝑃𝐢𝐡 are equal for the same reason. This time their shared arc is the arc 𝐴𝐡. We can therefore say that triangle 𝐴𝑃𝐷 and triangle 𝐡𝑃𝐢 are similar as all three angles are equal.

In order to work out the length 𝐡𝐢, we firstly need to find the scale factor. As triangle 𝐡𝑃𝐢 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 𝐡𝑃𝐢 is a third of the size of triangle 𝐴𝑃𝐷. In order to calculate the length of 𝐡𝐢 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 𝐡𝐢 is 19 over four or nineteen quarters. This could be rewritten as a decimal as 4.75.

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