Trigonometry & Inverse Trigonometry
Heights and Distances
Grade None
Question:
<p>\(PQR\) is a triangular park with \(PQ = PR = 200\) m. A TV tower stands at the mid-point of \(QR\). If the angles of elevation of the top of the tower at \(P\), \(Q\) and \(R\) are, respectively, 45°, 30° and 30°, then the height of the tower (in m) is:</p>
<p>50</p>
<p>\(100\sqrt{3}\)</p>
<p>\(50\sqrt{3}\)</p>
<p>100</p>
Step-by-Step Solution
Key Concept: Since PQR is isosceles with PQ = PR and the tower is at the midpoint M of QR, use the property that PM ⊥ QR. Set up two separate right triangles: one from P looking down at the tower top, and one from Q (or R) looking up at the tower top.
<p><strong>Step 1:</strong> Since PQ = PR = 200 m and M is the midpoint of QR, by isosceles triangle properties, PM ⊥ QR.</p><p><strong>Step 2:</strong> Let the tower height be h (from ground at M to top T). Let PM = d (height of P above ground).</p><p><strong>Step 3:</strong> From point Q, angle of elevation to T is 30°. If QM = x, then: tan(30°) = h/x, so x = h/√3 = h√3/3</p><p><strong>Step 4:</strong> From point P, angle of elevation to T is 45°. Since P is at height d above ground and T is at height h, the angle of elevation means: tan(45°) = (d - h)/PM = (d - h)/d. Therefore: 1 = (d - h)/d, giving d - h = d, which is impossible. Re-interpret: angle of elevation 45° means tan(45°) = h/(d) when looking from P down along PM to the base, adjusted for tower position.</p><p><strong>Step 5:</strong> Correct interpretation: tan(45°) = h/d ⟹ d = h</p><p><strong>Step 6:</strong> In right triangle PMQ: PM² + QM² = PQ². So d² + x² = 200². Substituting d = h and x = h√3:</p><p>h² + 3h² = 40000</p><p>4h² = 40000</p><p>h² = 10000</p><p>h = 100 m</p><p>∴ Answer: D (100 m)</p>
Correct Answer: D