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In triangle ABC, the measure of ∠B is 90°, BC = 16, and AC = 20. Triangle DEF is similar to triangle ABC, where vertices D, E, and F correspond to vertices A, B, and C, respectively, and each side of triangle DEF is 1/3 the length of thecorresponding side of triangle ABC. What is the value of sin F?​

Sagot :

SalmonWorldTopperidn

Step-by-step explanation:

[tex]\large\underline{\sf{Solution-}}[/tex]

Given that

[tex]\rm :\longmapsto\:\triangle ABC \: \sim \: \triangle DEF[/tex]

[tex]\bf\implies \:\dfrac{AB}{DE} = \dfrac{BC}{EF} = \dfrac{AC}{DF} [/tex]

and

[tex]\rm\implies \:\angle A = \angle D[/tex]

[tex]\rm\implies \:\angle B = \angle E[/tex]

[tex]\rm\implies \:\angle C = \angle F[/tex]

Also, given that,

[tex]\rm\implies \:\dfrac{DE}{AB} = \dfrac{DF}{AC} = \dfrac{EF}{BC} = \dfrac{1}{3} [/tex]

[tex]\rm :\longmapsto\:BC = 16 \: units[/tex]

[tex]\rm :\longmapsto\:AC = 20 \: units[/tex]

So, using Pythagoras Theorem, we have

[tex]\rm :\longmapsto\: {AC}^{2} = {AB}^{2} + {BC}^{2} [/tex]

[tex]\rm :\longmapsto\: {20}^{2} = {AB}^{2} + {16}^{2} [/tex]

[tex]\rm :\longmapsto\: 400 = {AB}^{2} + 256[/tex]

[tex]\rm :\longmapsto\: {AB}^{2} = 400 - 256[/tex]

[tex]\rm :\longmapsto\: {AB}^{2} = 144[/tex]

[tex]\bf\implies \:AB = 12 \: units[/tex]

Now,

[tex]\rm :\longmapsto\:sinC = \dfrac{AB}{AC} [/tex]

[tex]\rm :\longmapsto\:sinC = \dfrac{12}{20} [/tex]

[tex]\rm :\longmapsto\:sinC = \dfrac{3}{5} [/tex]

Now, As we have

[tex]\rm :\longmapsto\:\angle C = \angle F[/tex]

[tex]\rm :\longmapsto\:sin C = sin F[/tex]

[tex]\bf\implies \:sinF = \dfrac{3}{5} [/tex]

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EXPLORE MORE:-

1. Pythagoras Theorem :-

This theorem states that : In a right-angled triangle, the square of the longest side is equal to sum of the squares of remaining sides.

2. Converse of Pythagoras Theorem :-

This theorem states that : If the square of the longest side is equal to sum of the squares of remaining two sides, angle opposite to longest side is right angle.

3. Area Ratio Theorem :-

This theorem states that :- The ratio of the area of two similar triangles is equal to the ratio of the squares of corresponding sides.

4. Basic Proportionality Theorem :-

This theorem states that :- If a line is drawn parallel to one side of a triangle, intersects the other two lines in distinct points, then the other two sides are divided in the same ratio.

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