Question

12 fill in the blank 1 point use the circle below to find each measure given that hk = 30 and pm = 8. ph = type your answer... pn = type your answer... mg = type your answer...

Explanation:

Step1: Analyze PH (radius)

HK is a chord, and PN is perpendicular to HK (assuming from the diagram that PN bisects HK). So HN = HK/2 = 30/2 = 15? Wait, no, PH is a radius. Wait, maybe HK is a chord, and PM is the distance from center P to chord GL? Wait, maybe the diagram shows that HK is a chord, and PN is the perpendicular from center P to HK, so HN = NK = 15. But PH is the radius. Wait, maybe PM is the distance from center to chord GL, so triangle PMG is right-angled. Wait, let's clarify:

Assuming P is the center, so PH, PL, PK, PG are radii. HK is a chord, PN is perpendicular to HK, so HN = HK/2 = 15. PM is 8, which is the distance from center P to chord GL, so GL is a chord, PM ⊥ GL, so MG = ML, and by Pythagoras, MG = √(PH² - PM²). Wait, but we need to find PH first. Wait, maybe HK is a chord, and PN is perpendicular to HK, so triangle PHN is right-angled, with HN = 15, PN is... Wait, maybe the problem has HK as a chord, length 30, and PM = 8 (distance from center to chord GL). Wait, maybe PH is the radius, and we can find PH using HK? Wait, no, if PN is perpendicular to HK, then HN = 15, and if PN is, say, equal to PM? No, maybe the diagram shows that HK and GL are chords, PM and PN are perpendicular distances from center to chords. Wait, maybe PH is the radius, so let's assume that HK is a chord, length 30, and PN is the distance from center to HK, but we don't know PN. Wait, maybe the problem has a typo, or maybe I misinterpret. Wait, maybe PM is 8, and PH is the radius, so MG is calculated as √(PH² - PM²). But we need to find PH. Wait, maybe HK is a diameter? No, HK is 30, so if HK were a diameter, PH would be 15, but PM is 8. Wait, that makes sense! If HK is a diameter, then PH is the radius, so PH = 30/2 = 15. Then, since PM is the distance from center P to chord GL, and PM = 8, then in right triangle PMG, MG = √(PH² - PM²) = √(15² - 8²) = √(225 - 64) = √161? No, that can't be. Wait, no, maybe PM is the distance, and MG is half the chord length? Wait, no, chord length formula: length = 2√(r² - d²), where d is distance from center. So if GL is the chord, length GL = 2MG, and MG = √(r² - d²), where d = PM = 8, r = PH = 15. So MG = √(15² - 8²) = √(225 - 64) = √161 ≈ 12.69, but that seems odd. Wait, maybe I made a mistake. Wait, maybe HK is a chord, and PN is the distance, but PH is the radius. Wait, maybe the problem is that HK is a diameter? No, 30 would be diameter, so radius 15. Then PM is 8, so MG = √(15² - 8²) = √161 ≈ 12.69, but maybe the numbers are such that it's a whole number. Wait, maybe PM is 8, and PH is 17? Then 17² - 8² = 289 - 64 = 225, so MG = 15. Ah! That makes sense. So maybe HK is not a diameter, but PH is 17, HN is 15 (since 17² - 8²? No, wait, 15² + 8² = 225 + 64 = 289 = 17². So if HN is 15, PN is 8, then PH is 17. So HK is 30 (2*15), PN is 8, so PH = 17. Then, since PM is also 8 (distance from center to chord GL), then MG = √(PH² - PM²) = √(17² - 8²) = √(289 - 64) = √225 = 15. Wait, that works! So:

Step1: Find PH (radius)

HK is a chord, PN ⊥ HK, so HN = HK/2 = 30/2 = 15. PN = 8 (assuming PN is the distance from center to HK, same as PM? Wait, maybe PM and PN are both 8, so triangle PHN is right-angled with HN=15, PN=8, so PH = √(15² + 8²) = √(225 + 64) = √289 = 17. So PH = 17.

Step2: Find PN

Since PN is the distance from center to chord HK, and PM is the distance to chord GL, and if the chords are congruent or something, but actually, PN should be equal to PM? No, wait, maybe PN is also 8, but in triangle PHN, HN=15, PN=8, so PH=17. So PN is 8? Wait, no, PN is the segment from P to N, which is the foot on HK. So PN is 8? Wait, maybe the problem has PM=8, and PN=8, so both distances from center to chords are 8, so chords HK and GL are congruent, length 30 each? No, HK is 30, so GL is also 30? No, MG is half of GL, so MG=15. Wait, let's re-express:

- PH is radius: PH = 17 (from 15-8-17 triangle)
- PN: since PN is perpendicular to HK, and HN=15, PH=17, so PN = √(PH² - HN²) = √(17² - 15²) = √(289 - 225) = √64 = 8. So PN=8.
- MG: since PM=8 (distance from center to chord GL), and GL is a chord, PM ⊥ GL, so MG = √(PH² - PM²) = √(17² - 8²) = √(289 - 64) = √225 = 15.

So:

PH = 17, PN = 8, MG = 15.

Answer:

PH = 17
PN = 8
MG = 15