Please Help! Answer full step by step. Triangle ABC has vertices A(- 3, - 4), B(16, - 2) and C(13, - 10) . Show algebraically that ABC is a right angled triangle.

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Triangle ABC has the three sides AB, BC, and AC.

Let's find the slope of each line. We'll use the slope formula

Let's start with the slope of line AB.

A = (-3, -4) = (x1,y1)

B = (16, -2) = (x2,y2)

m = slope

m = (y2-y1)/(x2-x1)

m = (-2-(-4))/(16-(-3))

m = (-2+4)/(16+3)

m = 2/19

The slope of line AB is 2/19. I'll keep it in fraction form.

We'll use this later, so let p = 2/19.

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Repeat those steps to find the slope of BC

B = (16, -2) = (x1,y1)

C = (13, -10) = (x2,y2)

m = (y2-y1)/(x2-x1)

m = (-10-(-2))/(13-16)

m = (-10+2)/(13-16)

m = -8/(-3)

m = 8/3

The slope of line BC is 8/3.

Let q = 8/3 so we can use it later.

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Now find the slope of line AC.

A = (-3, -4) = (x1,y1)

C = (13, -10) = (x2,y2)

m = (y2-y1)/(x2-x1)

m = (-10-(-4))/(13-(-3))

m = (-10+4)/(13+3)

m = -6/16

m = -3/8

The slope of line AC is -3/8

Let r = -3/8.

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To recap everything so far, we have

We can see that,

Of those slope multiplications, q and r multiply to -1 is what we should focus on. If two slopes multiply to -1, then their lines are perpendicular. This applies to any perpendicular lines where neither line is vertical, and neither is horizontal.

The fractions 8/3 and -3/8 are negative reciprocals of one another. We flip the fraction and flip the sign from positive to negative, or vice versa. This is another way to see we have perpendicular slopes.

So in short, we've shown that BC and AC are perpendicular, and therefore triangle ABC is a right triangle.

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To solve this problem a completely different way, we could follow this outline:

Through step 1, you should find that,

Let a = sqrt(73), b = sqrt(292), c = sqrt(365). You should find that a^2+b^2 = c^2 is a true statement based on those values, and therefore we have a right triangle. The longest side is AB = sqrt(365), making angle C to be the 90 degree angle. The diagram is shown below. I used GeoGebra to make the diagram.

do you mean add both of the equations together

because if you do the answer is 10x-5

step-by-step explanation: