the vertices of a triangle are A(m,n) B (12,19) and C (23,20) where m and n are integers. If its area is 70 and slope of the median through A is -5 then the last digit of (m+n) is?
Answers
Given : the vertices of a triangle are A(m,n) B (12,19) and C (23,20) where m and n are integers. Area is 70 and slope of the median through A is -5
To find : the last digit of (m+n) is
Solution:
A(m,n) B (12,19) and C (23,20)
Mid point of BC
= (12 + 23)/2 , ( 19 + 20)/2
= 35/2 , 39/2
Slope of AD = (39/2 - n)/(35/2 - m) = - 5
=> (39 - 2n) = -5(35 - 2m)
=> 39 - 2n = -175 + 10m
=> 10m + 2n = 214
=> 5m + n = 107
A(m,n) B (12,19) and C (23,20)
Area = (1/2) | m(19 - 20) + 12(20 - n) + 23(n - 19) | = 70
=> | -m + 240 -12n + 23n - 437 | = 140
=> | -m + 11n - 197 | = 140
=> -m + 11n - 197 = ± 140
Case 1 : -m + 11n - 197 = 140
=> -m + 11n = 337
5m + n = 107
n = 32 , m = 15
Case 2 : -m + 11n - 197 = -140
=> -m + 11n = 57
5m + n = 107
n = 7 , m = 20
m + n = 32 + 15 = 47 or 7 + 20 = 27
Last Digit of m + n = 7
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