Question

Help? Dont understand how to figure this out

Answer 1

`|AB|^2 + |AC|^2 = 738 = |BC|^2`

`m_(AB) * m_(AC) = -1` so `AB` is perpendicular to `AC`

`Area = 184.5 " units"^2`

Explanation:

Given: Triangle with vertices `A(6,-33), B(21, -21), C(-6, -18)`

A) Show Converse of the Pythagorean Theorem that the triangle is a right triangle:

Find the length of the sides using the distance formula: `sqrt((x_2-x_1)^2 + (y_2-y_1)^2)`

`|AB| = sqrt((21-6)^2 + (-21--33)^2) = sqrt(15^2 + 12^2) = sqrt(369)= sqrt(9*41) = 3 sqrt(41)~~ 19.21`

`|AC| = sqrt((-6-6)^2 + (-18--33)^2) = sqrt((-12)^2 + 15^2) = sqrt(369)= sqrt(9*41) = 3 sqrt(41)~~ 19.21`

`|BC| = sqrt((-6-21)^2 + (-18--21)^2) = sqrt( (-27)^2+(-3)^2) = sqrt(738)= sqrt(9*82) = 3sqrt(81)`

Pythagorean Theorem: `a^2 + b^2 = c^2` for a right triangle:

`|AB|^2 + |AC|^2 = 369 + 369 = 738`

`|BC|^2 = 738`

B) Use slopes to show the triangle is a right triangle. Two of the slopes must be negative reciprocals. `Slope = (y_2-y_1)/(x_2-x_1)`

`m_(AB) = (-21 - -33)/(21-6) = 12/15 = 4/5`

`m_(AC) = (-18 - -33)/(-6-6) = 15/(-12) = -5/4`

Negative reciprocals are slopes that when multiplied `= -1:`

`m_(AB) * m_(AC) = 4/5 * -5/4 = -20/20 = -1`

C) Find the Area of the triangle:`" "Area = 1/2 bh`.

The longest side is the hypotenuse. This means one of the other sides is the base and the other side is the height. Since both are equal in length, it doesn't matter which is which.

`Area = 1/2 * sqrt(369) * sqrt(369) = 1/2 * 369 = 184.5 " units"^2`