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The diagram shows a triangle with sides of length $x$ cm, 11 cm and 13 cm and an angle of 80° - HSC - SSCE Mathematics Standard - Question 17 - 2019 - Paper 1

Question 17

The diagram shows a triangle with sides of length $x$ cm, 11 cm and 13 cm and an angle of 80°. Use the cosine rule to calculate the value of $x$, correct to two sig... show full transcript

Worked Solution & Example Answer:The diagram shows a triangle with sides of length $x$ cm, 11 cm and 13 cm and an angle of 80° - HSC - SSCE Mathematics Standard - Question 17 - 2019 - Paper 1

Step 1

Use the cosine rule

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Answer

To find the value of $x$ , we use the cosine rule, which states that for any triangle with sides $a$ , $b$ , and $c$ and an angle $heta$ opposite side $c$ :

$c^2 = a^2 + b^2 - 2ab \cdot \cos(\theta)$

In this case, let:

$a = 11$ cm
$b = 13$ cm
$heta = 80°$ (angle opposite side $x$ )
$c = x$ cm

Substituting these values into the formula gives us:

$x^2 = 11^2 + 13^2 - 2 \cdot 11 \cdot 13 \cdot \cos(80°)$

Step 2

Calculate $x$

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Answer

Substituting the values into the equation:

$x^2 = 121 + 169 - 2 \cdot 11 \cdot 13 \cdot \cos(80°)$

Calculating the cosine:

$\cos(80°) \approx 0.173648$

Thus,

$x^2 = 121 + 169 - 2 \cdot 11 \cdot 13 \cdot 0.173648$ $= 121 + 169 - 2 \cdot 11 \cdot 13 \cdot 0.173648$ $= 240 - 40.278$ $= 240.366...$

Taking the square root gives us: $x \approx \sqrt{240.366...} \approx 15.5027...$

Rounding to two significant figures, the value of $x$ is: $x \approx 16$

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