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6. Aboat sails due East from a harbour. A lighthouse is 500 metres due North of the harbour. When the boat is at position A it is 600 metres away from the lighthouse. It sails a further 400 metres to position B. Calculate the direct distance from position B to the lighthouse as shown by the dotted line. Do not use a scale drawing.

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6. Aboat sails due East from a harbour. A lighthouse is 500 metres due North of the harbour. When the boat is at position A it is 600 metres away from the lighthouse. It sails a further 400 metres to position B. Calculate the direct distance from position B to the lighthouse as shown by the dotted line. Do not use a scale drawing.

6. Aboat sails due East from a harbour. A lighthouse is 500 metres due North of the harbour. When the boat is at position A it is 600 metres away from the lighthouse. It sails a further 400 metres to position B. Calculate the direct distance from position B to the lighthouse as shown by the dotted line. Do not use a scale drawing.

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TheaElite · Tutor for 8 years

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1118.03 meters<br />The direct distance from position B to the lighthouse is approximately 1118.03 meters.

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## Step 1: Identify the problem<br />The problem involves calculating the direct distance from position B to the lighthouse. This is a right triangle problem where we know the lengths of two sides and we need to find the length of the hypotenuse.<br /><br />## Step 2: Apply the Pythagorean theorem<br />The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be written as:<br /><br />### \(c^2 = a^2 + b^2\)<br /><br />where:<br />- \(c\) is the length of the hypotenuse,<br />- \(a\) and \(b\) are the lengths of the other two sides.<br /><br />## Step 3: Substitute the known values into the formula<br />In this problem, the boat sails 600 meters from the harbor to position A, then sails a further 400 meters to position B. The lighthouse is 500 meters due north of the harbor. Therefore, we can substitute \(a = 500\) and \(b = 600 + 400 = 1000\) into the Pythagorean theorem:<br /><br />### \(c^2 = 500^2 + 1000^2\)<br /><br />## Step 4: Calculate the square of the hypotenuse<br />Calculate the square of the hypotenuse by adding the squares of the other two sides:<br /><br />### \(c^2 = 250000 + 1000000 = 1250000\)<br /><br />## Step 5: Find the length of the hypotenuse<br />The length of the hypotenuse is the square root of the square of the hypotenuse. Therefore, calculate the square root of 1250000:<br /><br />### \(c = \sqrt{1250000} = 1118.03\)
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