Tanθ in a right angle triangle is the ratio of its perpendicular to its base. The ratio that represents the Tan(∠H) is [tex]\dfrac{8}{15}[/tex].
What is Tangent (Tanθ)?
The tangent or tanθ in a right angle triangle is the ratio of its perpendicular to its base. it is given as,
[tex]\rm{Tangent(\theta) = \dfrac{Perpendicular}{Base}[/tex]
where,
θ is the angle,
Perpendicular is the side of the triangle opposite to the angle θ,
Base is the adjacent smaller side of the angle θ.
As it is given that the two of the given triangles are similar triangles, therefore, their corresponding sides will be in the same ratio.
[tex]\dfrac{QR}{HJ} = \dfrac{RP}{JG} = \dfrac{PQ}{GH}[/tex]
Further, the sides can be written as,
[tex]\dfrac{QR}{HJ} = \dfrac{RP}{JG}\\\\\dfrac{QR}{RP} = \dfrac{HJ}{JG}=\dfrac{15}{8}[/tex]
Now, the Tan(∠H) can be written as,
[tex]\rm{Tangent(\theta) = \dfrac{Perpendicular}{Base}[/tex]
Substituting the value for the ∠H in ΔHJG,
[tex]\rm Tangent(\angle H) = \dfrac{JG}{HJ}[/tex]
As the ratio is already known, therefore,
[tex]\rm Tangent(\angle H) = \dfrac{8}{15} = 0.5\bar3[/tex]
Hence, the ratio that represents the Tan(∠H) is [tex]\dfrac{8}{15}[/tex].
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