Two circles with centers O and Q, and radii r = 8 cm. The distance between centers OQ = 12 cm. The common chord AB intersects OQ at X. Find the length of the common chord AB.

Haya! Hapa kuna masuluhisho ya maswali yako:

Swali la 24:

a) Urefu wa kamba ya kawaida AB.
Kutokana na data, tuna duara mbili zenye vituo O na Q, na radii $r = 8$ cm. Umbali kati ya vituo $OQ = 12$ cm. Kamba ya kawaida AB hukutana na OQ kwa X. OQ ni bisector ya perpendicular ya AB.
Katika $\triangle OAX$, $OA = 8$ cm. Wacha $OX = x$ na $AX = y$.
Katika $\triangle QAX$, $QA = 8$ cm. Wacha $QX = 12 - x$.
Kutumia theorem ya Pythagoras:
Katika $\triangle OAX$:

$$OA^2 = OX^2 + AX^2$$ $$8^2 = x^2 + y^2$$ $$64 = x^2 + y^2 \quad \text{(i)}$$

Katika $\triangle QAX$:

$$QA^2 = QX^2 + AX^2$$ $$8^2 = (12-x)^2 + y^2$$ $$64 = (12-x)^2 + y^2 \quad \text{(ii)}$$

Kutoa (i) kutoka (ii):

$$0 = (12-x)^2 - x^2$$ $$0 = (144 - 24x + x^2) - x^2$$ $$0 = 144 - 24x$$ $$24x = 144$$ $$x = \frac{144}{24} = 6 \text{ cm}$$

Sasa badilisha $x=6$ katika (i):

$$64 = 6^2 + y^2$$ $$$$