Question

9. A rectangle has dimensions 10 cm by 5
cm. Determine the measures of the angles at
the point where the diagonals intersect.
10. The lengths of side AB and side BC of a
scalene triangle ABC are 12 cm and 8 cm
respectively. The size of angle C is 59º. Find
the length of side AC.

Answer 1

$\huge\colorbox{pink}{answer} \: \blacktriangledown$

$\bold{ \star \: \angle \: a = 12 \: cm} \\ \bold{ \star \: \angle \: b = 8 \: cm} \\ \bold{\star \: \angle \: c = 59 \degree ≈ \: 60 \degree}$

$\red{\underline{ \underline\bold{\dagger \: using \: consine \: formulae}}}$

$\bold{ \leadsto \: \frac{\cos(c) \: = \: {a}^{2} + {b}^{2} - {c}^{2}}{2ab}}$

$\sf\leadsto \cos(60 \degree) = \frac{ {12}^{2} + {8}^{2} - {c}^{2} }{2 \times 12 \times 8}$

$\sf \leadsto \: \frac{1}{2} \: = \: \frac{144 + 64 - {c}^{2} }{196}$

$\leadsto \sf \frac{1}{2} = \frac{208 - {c}^{2} }{192}$

$\sf \leadsto \: 416 - 2 {c}^{2} = 192$

$\sf \leadsto \: {2c}^{2} = 416 - 192$

$\sf \leadsto \: {2c}^{2} = 224$

$\sf \leadsto \: {c}^{2} = 112$

$\sf \leadsto \: c = \sqrt{112} = 10.58$

$\sf\leadsto \: c = 10.58 \: ≈ \: 10$

$\hookrightarrow \fbox{\underline{\: c = 10 \: cm}}$

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Answer 2

Step-by-step explanation:

$\huge\colorbox{pink}{answer} \: \blacktriangledown$

$\bold{ \star \: \angle \: a = 12 \: cm} \\ \bold{ \star \: \angle \: b = 8 \: cm} \\ \bold{\star \: \angle \: c = 59 \degree ≈ \: 60 \degree}$

$\red{\underline{ \underline\bold{\dagger \: using \: consine \: formulae}}}$

$\bold{ \leadsto \: \frac{\cos(c) \: = \: {a}^{2} + {b}^{2} - {c}^{2}}{2ab}}$

$\sf\leadsto \cos(60 \degree) = \frac{ {12}^{2} + {8}^{2} - {c}^{2} }{2 \times 12 \times 8}$

$\sf \leadsto \: \frac{1}{2} \: = \: \frac{144 + 64 - {c}^{2} }{196}$

$\leadsto \sf \frac{1}{2} = \frac{208 - {c}^{2} }{192}$

$\sf \leadsto \: 416 - 2 {c}^{2} = 192$

$\sf \leadsto \: {2c}^{2} = 416 - 192$

$\sf \leadsto \: {2c}^{2} = 224$

$\sf \leadsto \: {c}^{2} = 112$

$\sf \leadsto \: c = \sqrt{112} = 10.58$

$\sf\leadsto \: c = 10.58 \: ≈ \: 10$

$\hookrightarrow \fbox{\underline{\: c = 10 \: cm}}$

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