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Math Chapter 6 : Topic 2
Pythagoras and Trig Ratios
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In a right-angled triangle, if two sides are $6 cm$ and $8 cm$, the hypotenuse is:
Reasoning: ${6}^{2}+{8}^{2}=36+64=100 \Longrightarrow \sqrt{100}=10$Pythagorean theorem:${(Perpendicular)}^{2}+{(Base)}^{2}={(hypotenuse)}^{2}$
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In a right triangle, if hypotenuse $=13 cm$ and one side $=5 cm$, the other side is:
Reasoning: ${13}^{2}-{5}^{2}=169-25=\sqrt{144} \Longrightarrow 12$Pythagorean theorem:${(Perpendicular)}^{2}+{(Base)}^{2}={(hypotenuse)}^{2}$
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If $\theta=$ $\frac{3}{5} $, the triangle is:
Reasoning: $\frac{Opp}{hyp}=\frac{3}{5} \Longrightarrow$ adjacent $=4$
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In a right triangle, if opposite $=4 cm$ and adjacent $=3 cm$, then $tan \theta$ is:
Reasoning: $tan=$ $\frac{opp}{adj}$ $=$ $\frac{4}{3}$
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If $cos \theta=$ $\frac{12}{13} $, then $sin \theta$ is:
Reasoning: Triangle$ \Longrightarrow 5-12-13$
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In a triangle, if one side is $9 cm$, another is $12 cm$, the hypotenuse is:
Reasoning: ${9}^{2}+{12}^{2}=81+144=225 \Longrightarrow 15$
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If $tan \theta=1$, then $\theta$ is:
Reasoning: $Opp=adj \Longrightarrow {45}^{o}$
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If hypotenuse $=10 cm$ and adjacent $=6 cm$, then $cos \theta$ is:
Reasoning: $6/10=3/5$
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If opposite $=5$ and hypotenuse $=13$, then adjacent is:
Reasoning: ${13}^{2}-{5}^{2}=\sqrt{144} \Longrightarrow 12$
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If $sin \theta=$ $\frac{1}{2}$ then $\theta$ is:
Reasoning: Standard value.
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A ladder $13 m$ long reaches a wall. If base is $5 m$ from wall, height is:
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If $cos \theta=$ $\frac{4}{5} $, then $tan \theta$ is:
Reasoning: Triangle $3-4-5 \Longrightarrow tan=3/4$
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If adjacent $=7 cm$ and opposite $=24 cm$, hypotenuse is:
Reasoning: ${7}^{2}+{24}^{2}=\sqrt{625} \Longrightarrow 25$
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Which ratio is used to find height when base and angle are known?
Reasoning: $tan=opp / adj$
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If $sin \theta=$ $\frac{5}{13} $, then hypotenuse is:
Reasoning: $sin=opp / hyp \Longrightarrow hyp=13$
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A pole casts a shadow $10 m$ long. If height is $10 m$, angle is:
Reasoning: $tan=$ $\frac{10}{10}=1 \Longrightarrow {45}^{o}$
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If hypotenuse $=17$ and one side $=8$, other side is:
Reasoning: ${17}^{2}-{8}^{2}=225 \Longrightarrow \sqrt{225} \Longrightarrow 15$
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If $tan=$ $\frac{3}{4} $, then $sin \theta$ is:
Reasoning: $3-4-5$ triangle$ \Longrightarrow sin=3/5$
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If a triangle has sides $5$, $12$, $13$, the angle opposite $12$ is:
Reasoning: Right angle is opposite $13$, so $12$ corresponds to acute.
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Which ratio equals $\frac{adjacent}{hypotenuse} $?
Reasoning: Definition of cosine.
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If $cos \theta=$ $\frac{8}{17} $, then opposite is:
Reasoning: $8-15-17$ triangle.
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Which method is used when only sides are given?
Reasoning: Pythagoras uses side lengths.
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If height $=6 m$ and hypotenuse $=10 m$, base is:
Reasoning: ${10}^{2}-{6}^{2}=\sqrt{64} \Longrightarrow 8$
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If $sin \theta=$ $\frac{4}{5} $, then $cos \theta$ is:
Reasoning: $3-4-5$ triangle.
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Which relation connects all three sides of a right triangle?
Reasoning: Pythagoras theorem.
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