Questions: Example 2: Missing Data: Lengths of Two Sides

Example 2: Missing Data: Lengths of Two Sides
Transcript text: Example 2: Missing Data: Lengths of Two Sides
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Solution

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Solution Steps

Step 1: Analyze the given information

We are given a quadrilateral ABCD. We know the length of AB = 610.67 and the measures of the following angles: ∠DAB = 83°46' ∠ABC = 92°38' ∠CDA = 46°21' ∠ADB = 70°8.01', so ∠ADC = 70°8.01' + 46°21' = 116°29.01' We also know the length of AD = 512.57 and the length of CD = 462.1

We are looking for the lengths of BC and CD. We are given the length of CD so we are looking for BC.

Step 2: Solve for the length of BC

We are given CD. Since we know two sides (AD and AB) and the angle between them (∠DAB), we can solve for the length of BD using the law of cosines. For triangle ABD: BD² = AD² + AB² - 2 * AD * AB * cos(∠DAB) BD² = 512.57² + 610.67² - 2 * 512.57 * 610.67 * cos(83°46') BD² = 262716.3049 + 372926.9489 - 625452.7938 * cos(83.7667°) BD² ≈ 262716.30 + 372926.95 - 625452.79 * 0.1111 BD² ≈ 635643.25 - 69478.55 BD² ≈ 566164.7 BD ≈ 752.44

Now, in triangle BCD we know the lengths of BD and CD and the angle between them, so we can use the law of cosines again:

BC² = BD² + CD² - 2 * BD * CD * cos(∠BDC) BC² = 752.44² + 462.1² - 2 * 752.44 * 462.1 * cos(46°21') BC² = 566168.3536 + 213540.41 - 695131.324 * 0.6914 BC² ≈ 779708.76 - 480294.87 BC² ≈ 299413.89 BC ≈ 547.19

Final Answer:

The length of BC is approximately 547.19.

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