Question

A circular lake is located at a certain barangay in Tagaytay. An existing road runs tangential to the circular lake. Twelve kilometers from the point of tangency, along the existing road, a new road of 8 km length is constructed to the periphery of the lake at B. The new road is extended to cross the lake. Compute the length of the bridge needed to cross the lake

Answer 1

First we sketch the situation as done below

P 8km B 12km e ? е С. ΑΙ o

Given that the sand existing soad AP is tangential to the circular lake. Length AP = 12 km From point p a road (new) of 8 km length is constructed to the periphery of the lake at B. so PB = 8 km. The new road is constan entended to cross the lake. Let the bridge be Bc. So to find BC= ? Let TBAP=0. Also angle SOAP = 90° as In the AP is tangent, so OAL AP In the triangle, AOAB, OA= OB=radius of Circl. COAB=<OB A = <OAP - <BAP = 90°-o Also AOB = 180°- (<OAB +SOBA) 2180 - (90-0 +90-0) Then <BOC = 180° - LAOB = 180° -20 Thug <OBC=C0CB = 0

Now observe that <ABOTSOBC = (90-0) +0 = 90° = (ABC CABP = 180 - <ABC A ABP = 180-90°-90° right angled triangle. Thus, ✓ 144564 Now A AP=12, BP = 8 =) AB=√12²2² os ſ80 = tan (BAP Also tamo BP AB Next 88 vso in the right angled triangle 4ABC, tam (CACB) = tem o= AB ✓so BC 2 8. =) BC= B C 8 Uso 800 - 10 8 Thus, the length of the bridge BC = 10km, (Am.

The answer is Bridge length is 10km.

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