Question

the hypotenuse of a isosceles right triangle lies along the line 2x-y=4 and the vertex opposite to hypotenuse is (1,5) .Obtain the other 2 sides.

Answer 1

Isosceles Right triangle ABC, < B = 90°, AB = BC & hypotenuse AC is 2x-y = 4 . Coordinates of B is ( 1,5)

Here < C= < A = 45°

● First i shall find out the slope of each side of the triangle ABC .

Slope of hypotenuse AC 2x - y = 4

= - ( coefficient of x)/ ( coefficient of y)

= -2/-1 = 2

● Now, find out Slope of equal sides BC & AB , using angle 45° & the slope of hypotenuse

Angle between AC & BC also angle between AC & AB = 45°

tan 45°=+,- (m - slope of AC)/(1+ m* slope of Ac)

Here, we take + & - both, as there are 2 equal sides making 45° with AC.

=> 1 =( m-2) / (1+ 2m)

=> 1+2m = m-2

=> m = -3 …………. (A)

Next, 1 = - ( m-2)/(1+2m)

=> 1+2m = -m+2

=> 3m = 1

=> m = 1/3 …………..(B)

● Now the equation of line ( say BC) passing through B(1,5), with slope -3

y - 5 = (x-1) *(-3)

=> y-5 = -3x + 3

=> 3x + y = 8 ……….. (1) represented by BC

● Next equation of line ( AB) passing through B(1,5) , with slope 1/3

y -5 = (x-1) * (1/3)

=> y - 5 = x/3 - 1/3

=> 3y - 15 = x-1

=> x- 3y = -14 ……………..(2) represented by AB

● Now find the coordinates of C

By solving 2x - y = 4 & 3x + y = 8 , we get x= 2.4 & y = 0.8

=> coordinates of C = ( 2.4, 0.8)

● Now find the coordinates of A

By solving 2x- y = 4 & x- 3y = -14, we get x= 5.2 & y = 6.4

=> cordinates of A = ( 5.2, 6.4)

& coordinates of B = (1, 5)

Now using section formula we get

AB = √( 4.2² + 1.4²) = 4.42 unit

BC = √( 1.4² + 4.2²) = 4.42 unit

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