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Right Triangle Calculation Puzzle: Solve for ABC, ADB, BDC

In this problem, we are given a right triangle ABC with a right angle at B, where BD is perpendicular to AC. Given CD=9 and AD=3, we find the exact value of BC+AB+BD using the Pythagorean theorem. By breaking down the triangle into ΔABC, ΔADB, and ΔBDC, and applying the theorem, we derive the lengths of AB, BC, and BD to find the solution. The final answer is √108 + √27 + 6 = 9√3 + 6.

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Right Triangle Calculation Puzzle: Solve for ABC, ADB, BDC

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  1. Problem Presentation set 2 problem 12 Ellen Dickerson

  2. Before we start on the problem lets review a little bit…. • Line segment • Distance of a line segment • Angle • Pythagorean theorem

  3. In the figure to the right, ΔABC is a right triangle, with a right angle at B, BD is perpendicular to AC. If CD=9, and AD=3, then find the exact value BC+AB+BD. AD=3 CD=9

  4. Lets start finding the three right triangles in the picture • ΔABC (the right angle is at ∠ ABC) • ΔADB (the right angle is at ∠ ADB) • ΔBDC (the right angle is at ∠ BDC) AD=3 CD=9

  5. Now we will use the Pythagorean Theorem (a2+b2=c2)to find a formula for each triangle • ΔABC: (AB)2 + (BC)2 = 122 • ΔADB: 32 + (DB)2 = (AB)2 • ΔBDC: 92 +(DB)2 = (BC)2 AD=3 CD=9

  6. ΔABC: (AB)2 + (BC)2 = 122 • ΔADB: 32 + (DB)2 = (AB)2 • ΔBDC: 92 + (DB)2 = (BC)2 • (AB)2 +(BC)2 = 122 • (AB)2 + [ 92 +(DB)2] = 122 • (AB)2 + 81 + (BD)2 = 144 • [32 + (BD)2] + 81 +(BD)2 = 144 • 90 + 2(BD)2 = 144 • 2(BD)2 = 54 • (BD)2 = 27 • BD = √27 AD=3 CD=9

  7. ΔABC: (AB)2 + (BC)2 = 122 • ΔADB: 32 + (DB)2 = (AB)2 • ΔBDC: 92 + (DB)2 = (BC)2 • 32 + (DB)2 = (AB)2 • 9 +27 = (AB)2 • 36 = (AB)2 • 6 = AB AD=3 BD=√27 CD=9

  8. ΔABC: (AB)2 + (BC)2 = 122 • ΔADB: 32 + (DB)2 = (AB)2 • ΔBDC: 92 + (DB)2 = (BC)2 • 92 + (DB)2 = (BC)2 • 81 + 27 = (BC)2 • 108 = (BC)2 • √108 = BC AD=3 BD=√27 CD=9 AB=6

  9. The problem asked us to find BC+BD +AB • √108 + √27 + 6 • √(36 *3) + √(9*3) + 6 • 6√3 + 3√3 + 6 • 9√3 + 6 • . AD=3 BD=√27 CD=9 AB=6 BC=√108

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