Question

The points A, B, and C have coordinates (2,1) , (b,3) , and (5,5), where b>3 and the angle of ABC is 90 degrees. Find the value of b, the lengths of AB and BC, and the area of triangle ABC!

Please answer, it means alotttt thanks :)

Answer 1

Answer:

sorry I can't understand

Answer 2

Answer:

Consider first the line from A to B

Where A is at x = 2 and y = 1

And B is at x = b and y = 3

The equation of this straight line will be of the form:

y = mx + c

Where m is the slope or gradient and c is a constant.

As it passes through A (2,1)

We know

1 = 2m + c

And we know it goes through B (b,3)

Hence we know

3 = bm + c

Subtract these two equations to eliminate c

3 - 1 = bm -2m

2 = m(b - 2)

Hence m = 2/ (b-2)

So the perpendicular line will be the inverse negative slope.

Ie the perpendicular from C will cross the first line at B ( b,3)

This equation will be of form

y = mx + c again but m and c will be different constants to the first line

y = - (b-2)/2 times x + d where m is the negative inverse and we call the new constant d to avoid confusion

This perpendicular passes through C (5,5)

Hence

5 = - (b-2)/2 times 5 + d

Multiply every term by 2 to get rid of fractions

10 = - 5(b-2) + 2d

10 = -5b + 10 + 2d

0 = -5b + 2d

5b = 2d

The perpendicular also passes through B (b,3)

Hence

3 = - (b-2)/2 times b + d

Multiply all terms by 2

6 = - (b-2)b +2d

But 2d = 5b

Hence

6 = - (b-2)b + 5b

6 = - b^2 + 2b +5b

6 = - b^2 + 7b

Collect all terms to left hand side (LHS)

b^2 - 7b + 6 = 0

(b-6)(b-1) = 0

Hence b = 6 or b = 1

We are told b is greater than 3

So b = 6

ANSWER

b = 6

We can check this is correct if we proceed a little

m = 2/(b-2)

= 2/(6-2)

= 2/4

= 1/2

Hence equation of first line is

y = x/2 + c

And we know the line passes through A (2,1)

So

1 = 1 + c

So

c = 0

First equation is y = x/2 + 0

Or 2y = x

The perpendicular to this line which meets at B is given by the equation:

y = - (b-2)/2 times x + d

We know it goes though C (5,5)

So

5 = - 4/2 times 5 + d

5 = -10 + d

So

d = +15

Hence the equation for the perpendicular is

y = - (b-2)/2 times x + d

y = -2x + 15

Collect all terms to LHS

y + 2x - 15 = 0

Step-by-step explanation: