Math  /  Geometry

QuestionPada lingkaran yang berpusat di O , titik P berada di luar lingkaran. PA dan PB garis singgung dengan panjang PA=20 cm\mathrm{PA}=20 \mathrm{~cm} dan PO=25 cm\mathrm{PO}=25 \mathrm{~cm}. panjang tali busur AB adalah...

Studdy Solution
Since PA PA and PB PB are tangents, PAO \triangle PAO and PBO \triangle PBO are right triangles. The line AB AB is a chord of the circle. The length of the chord can be found using the Pythagorean theorem in AOB \triangle AOB , where AO=BO=r AO = BO = r .
The length of the chord AB AB is given by:
AB=2×AO2AM2 AB = 2 \times \sqrt{AO^2 - AM^2}
where M M is the midpoint of AB AB , and AM=AO2OM2 AM = \sqrt{AO^2 - OM^2} .
Since OM OM is perpendicular to AB AB and OM=PO2PA2 OM = \sqrt{PO^2 - PA^2} :
OM=252202 OM = \sqrt{25^2 - 20^2} OM=625400 OM = \sqrt{625 - 400} OM=225 OM = \sqrt{225} OM=15cm OM = 15 \, \text{cm}
Since AO=15cm AO = 15 \, \text{cm} , and OM=15cm OM = 15 \, \text{cm} , AM=0 AM = 0 .
Therefore, AB=2×AO202=2×AO=2×15=30cm AB = 2 \times \sqrt{AO^2 - 0^2} = 2 \times AO = 2 \times 15 = 30 \, \text{cm} .
The length of the chord AB AB is:
30cm \boxed{30 \, \text{cm}}

View Full Solution - Free
Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord