Answer:
2. Because the triangles are similar, the cosine of angle C is the same as the cosine of angle T.
3. 0.86 = x/15
Step-by-step explanation:
Given the additional information that triangles [tex]\sf \triangle ABC [/tex] and [tex]\sf \triangle RST [/tex] are similar and the cosine of angle [tex]\sf C [/tex] is [tex]\sf 0.86 [/tex], let's analyze the statements:
For left triangle [tex]\sf \triangle ABC [/tex]:
Base: [tex]\sf BC = x [/tex]
Hypotenuse: [tex]\sf AC = 15 [/tex]
For right triangle [tex]\sf \triangle RST [/tex]:
Base: [tex]\sf ST = 28 [/tex]
Hypotenuse: [tex]\sf RT = y [/tex]
Given that [tex]\sf \triangle ABC [/tex] and [tex]\sf \triangle RST [/tex] are similar triangles, their corresponding angles are equal. Therefore, we have:
[tex]\sf \dfrac{BC}{ST} = \dfrac{AC}{RT} [/tex]
Substituting the given values:
[tex]\sf \dfrac{x}{28} = \dfrac{15}{y} [/tex]
Now, let's evaluate each statement:
[tex]\sf \sin T = \cos C [/tex]
In a right triangle, [tex]\sf \sin T = \dfrac{BC}{RT} = \dfrac{x}{y} [/tex] and [tex]\sf \cos C = \dfrac{BC}{AC} = \dfrac{x}{15} [/tex]. Therefore, this statement is false because [tex]\sf \sin T [/tex] and [tex]\sf \cos C [/tex] are not necessarily equal in this context.
Because the triangles are similar, the cosine of angle C is the same as the cosine of angle T.
This statement is true. In similar triangles, corresponding angles are equal, so [tex]\sf \cos C = \cos T [/tex].
[tex]\sf 0.86 = \dfrac{x}{15} [/tex]
Given that [tex]\sf \cos C = 0.86 [/tex] and [tex]\sf \cos C = \dfrac{x}{15} [/tex], this statement is true. Solving for [tex]\sf x [/tex]:
[tex]\sf \dfrac{x}{15} = 0.86 [/tex]
[tex]\sf x = 0.86 \times 15 [/tex]
[tex]\sf x = 12.9 [/tex]
[tex]\sf \dfrac{x}{15} = \dfrac{y}{15} [/tex]
Since [tex]\sf \dfrac{x}{28} = \dfrac{15}{y} [/tex], we know that [tex]\sf x [/tex] is to [tex]\sf 28 [/tex] as [tex]\sf 15 [/tex] is to [tex]\sf y [/tex].
Therefore, this statement is false unless further information is provided to establish a direct relationship between [tex]\sf x [/tex] and [tex]\sf y [/tex].
In summary:
Statement 1 is false.
Statement 2 is true.
Statement 3 is true.
Statement 4 is false unless additional information is provided.
Therefore, the true statements are 2 and 3.
