Question Number 220693 by hardmath last updated on 17/May/25
Commented by mr W last updated on 18/May/25
$${x}=\mathrm{6}? \\ $$
Commented by hardmath last updated on 18/May/25
$$ \\ $$Dear professor, I don't know the answer..
Answered by A5T last updated on 18/May/25
$$\mathrm{Let}\:\mathrm{D}\:\mathrm{be}\:\mathrm{the}\:\mathrm{point}\:\mathrm{separating}\:\mathrm{segments}\:\mathrm{with} \\ $$$$\mathrm{length}\:\:\mathrm{9}\:\mathrm{and}\:\mathrm{4} \\ $$$$\sqrt{\mathrm{R}^{\mathrm{2}} −\left(\mathrm{6}.\mathrm{5}\right)^{\mathrm{2}} }=\sqrt{\mathrm{CD}^{\mathrm{2}} −\mathrm{2}.\mathrm{5}^{\mathrm{2}} } \\ $$$$\Rightarrow\mathrm{CD}=\sqrt{\mathrm{R}^{\mathrm{2}} −\mathrm{36}} \\ $$$$\mathrm{R}^{\mathrm{2}} =\mathrm{CD}^{\mathrm{2}} +\mathrm{x}^{\mathrm{2}} \Rightarrow\mathrm{x}^{\mathrm{2}} =\mathrm{R}^{\mathrm{2}} −\left(\mathrm{R}^{\mathrm{2}} −\mathrm{36}\right) \\ $$$$\Rightarrow\mathrm{x}=\mathrm{6} \\ $$
Answered by mr W last updated on 18/May/25
Commented by mr W last updated on 18/May/25
$${AC}\:{is}\:{diameter}\:{of}\:{small}\:{circle}, \\ $$$$\Rightarrow\angle{ADC}=\mathrm{90}° \\ $$$${CB}={radius}\:{of}\:{big}\:{circle}={CA} \\ $$$$\Rightarrow{BD}={y}={AD}={x} \\ $$$$\frac{{x}}{\mathrm{4}}=\frac{\mathrm{9}}{{y}}\Rightarrow{xy}=\mathrm{9}×\mathrm{4}\:\Rightarrow{x}=\sqrt{\mathrm{9}×\mathrm{4}}=\mathrm{6}\:\checkmark \\ $$
Commented by hardmath last updated on 18/May/25
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{my}\:\mathrm{dear}\:\mathrm{professors}\:\mathrm{cool} \\ $$
Commented by fantastic last updated on 18/May/25
$${best}\:{solution} \\ $$$${It}\:{would}\:{be}\:{great}\:{if}\:{you}\:{add}\:'\:{chord}\:{intersection}\:\:{theorem}' \\ $$
Commented by mr W last updated on 18/May/25
$${i}\:{showed}\:{this}\:{with} \\ $$$$\frac{{x}}{\mathrm{4}}=\frac{\mathrm{9}}{{y}}\:\Rightarrow{xy}=\mathrm{9}×\mathrm{4} \\ $$
Answered by Spillover last updated on 18/May/25
Intersecting chord theorem
x*x=9*4
x²=36
x=6
x*x=9*4
x²=36
x=6