Question

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 the corresponding side of triangle ABC. What is the value of sinF?



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Answer 1

Answer:-

• 3/5 or 0.6

Explanation:-

Triangle ABC is a right triangle with its right angle at B. Therefore, AC is the hypotenuse of right triangle ABC, and AB and BC are the legs of right triangle ABC.

According to the Pythagorean theorem,

AB= √202−162

=√400−256

=√144 =12

Since triangle DEF is similar to triangle ABC, with vertex F corresponding to vertex C, the measure of angle∠F equals the measure of angle∠C. Therefore, sinF = sinC.

From the side lengths of triangle ABC,

sinF = opposite side/ hypotenuse

=AB / AC

=12 / 20

=3 / 5

Therefore, sinF = 3/5

The final answer is  3/5  or 0.6.

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Answer 2

ᴀɴꜱᴡᴇʀ ᴇxᴘʟᴀɴᴀᴛɪᴏɴ:

ᴛʀɪᴀɴɢʟᴇ ᴀʙᴄ ɪꜱ ᴀ ʀɪɢʜᴛ ᴛʀɪᴀɴɢʟᴇ ᴡɪᴛʜ ɪᴛꜱ ʀɪɢʜᴛ ᴀɴɢʟᴇ ᴀᴛ ʙ. ᴛʜᴇʀᴇꜰᴏʀᴇ, ᴀᴄ ɪꜱ ᴛʜᴇ ʜʏᴘᴏᴛᴇɴᴜꜱᴇ ᴏꜰ ʀɪɢʜᴛ ᴛʀɪᴀɴɢʟᴇ ᴀʙᴄ, ᴀɴᴅ ᴀʙ ᴀɴᴅ ʙᴄ ᴀʀᴇ ᴛʜᴇ ʟᴇɢꜱ ᴏꜰ ʀɪɢʜᴛ ᴛʀɪᴀɴɢʟᴇ ᴀʙᴄ. ᴀᴄᴄᴏʀᴅɪɴɢ ᴛᴏ ᴛʜᴇ ᴘʏᴛʜᴀɢᴏʀᴇᴀɴ ᴛʜᴇᴏʀᴇᴍ,

ᴀʙ=√202−162=√400−256=√144=12

ꜱɪɴᴄᴇ ᴛʀɪᴀɴɢʟᴇ ᴅᴇꜰ ɪꜱ ꜱɪᴍɪʟᴀʀ ᴛᴏ ᴛʀɪᴀɴɢʟᴇ ᴀʙᴄ, ᴡɪᴛʜ ᴠᴇʀᴛᴇx ꜰ ᴄᴏʀʀᴇꜱᴘᴏɴᴅɪɴɢ ᴛᴏ ᴠᴇʀᴛᴇx ᴄ, ᴛʜᴇ ᴍᴇᴀꜱᴜʀᴇ ᴏꜰ ᴀɴɢʟᴇ∠ꜰ ᴇQᴜᴀʟꜱ ᴛʜᴇ ᴍᴇᴀꜱᴜʀᴇ ᴏꜰ ᴀɴɢʟᴇ∠ᴄ. ᴛʜᴇʀᴇꜰᴏʀᴇ, ꜱɪɴꜰ=ꜱɪɴᴄ. ꜰʀᴏᴍ ᴛʜᴇ ꜱɪᴅᴇ ʟᴇɴɢᴛʜꜱ ᴏꜰ ᴛʀɪᴀɴɢʟᴇ ᴀʙᴄ,

ꜱɪɴꜰ = ᴏᴘᴘᴏꜱɪᴛᴇꜱɪᴅᴇ/ʜʏᴘᴏᴛᴇɴᴜꜱᴇ = ᴀʙ/ᴀᴄ = 12/20 = ⅗

ᴛʜᴇʀᴇꜰᴏʀᴇ, ꜱɪɴꜰ=⅗.

ᴛʜᴇ ꜰɪɴᴀʟ ᴀɴꜱᴡᴇʀ ɪꜱ ⅗ ᴏʀ 0.6.

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