Miscellaneous

Chapter 12 Class 11 - Intro to Three Dimensional Geometry (Term 2)
Serial order wise ### Transcript

Misc 3 If origin is the centroid of the triangle PQR with vertices P (2a, 2, 6), Q (–4, 3b, –10) and R (8, 14, 2c), then find the values of a, b and c. Given Δ PQR where P (2a, 2, 6) , Q (−4, 3b, –10) , R (8, 14, 2c) Also, Origin O (0, 0, 0) is the centroid of Δ PQR We know that Co ordinate of centroid whose vertices are (x1, y1, z1), (x2, y2, z2), (x3, y3, z3) is ((𝑥_1 + 𝑦_1 + 𝑧_1)/3,(𝑥_2 + 𝑦_2 + 𝑧_2)/3,(𝑥_3 + 𝑦_3 + 𝑧_3)/3) Here, x1 = 2a , y1 = 2 , z1 = 6 x2 = – 4 , y2 = 3b , z2 = –10 x3 = 8 , y2 = 14 , z3 = 2c ∴ Coordinates of centroid O(0, 0, 0) (0, 0, 0) = ((2𝑎 + (−4) + 8)/3,(2 + 3𝑏 + 14)/3,(6 + (−10) + 2𝑐)/3) (0, 0, 0) = ((2𝑎 − 4 + 8)/3,(2 + 3𝑏 + 14)/3,(6 − 10 + 2𝑐)/3) (0, 0, 0) = ((2𝑎 + 4)/3,(3𝑏 + 16)/3,(2𝑐 − 4)/3) x – coordinate 0 = (2𝑎 + 4)/3 3(0) = 2a + 4 0 = 2a + 4 2a + 4 = 0 2a = – 4 a = (−4)/2 a = –2 y – coordinate 0 = (3𝑏 + 16)/3 0(3) = 3b + 16 0 = 3b + 16 3b +16 = 0 3b = – 16 b = (−16)/3 z – coordinate 0 = (2𝑐 − 4)/3 3(0) = 2c – 4 0 = 2c – 4 2c – 4 = 0 2c = 4 c = 4/2 c = 2 Thus, a = – 2 , b = (−𝟏𝟔)/𝟑 & c = 2
{"ft_lang_label":"__label__en","ft_lang_prob":0.68555063,"math_prob":1.0000098,"size":1195,"snap":"2022-05-2022-21","text_gpt3_token_len":702,"char_repetition_ratio":0.109151974,"word_repetition_ratio":0.06583072,"special_character_ratio":0.65104604,"punctuation_ratio":0.17846154,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999658,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,4,null,4,null,4,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-23T14:51:57Z\",\"WARC-Record-ID\":\"<urn:uuid:63eeb41b-5e9c-4a26-a1a1-a8a3fa6b569d>\",\"Content-Length\":\"140346\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:24f57bb4-d330-422f-82b0-b726cd392e68>\",\"WARC-Concurrent-To\":\"<urn:uuid:e0c4fa6b-d008-44cc-9da6-b33ef9eb482a>\",\"WARC-IP-Address\":\"50.16.49.81\",\"WARC-Target-URI\":\"https://www.teachoo.com/2821/641/Misc-3---If-origin-is-centroid-of-PQR-with-P-(2a--2--6)/category/Miscellaneous/\",\"WARC-Payload-Digest\":\"sha1:NBDYMOOSLK2O3T5GBLW2UMSWZA5HUBPC\",\"WARC-Block-Digest\":\"sha1:AKRWZSJ5G635QL5TEY4WQFE2Y6ASLUDI\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652662558030.43_warc_CC-MAIN-20220523132100-20220523162100-00551.warc.gz\"}"}