Two corners of an isosceles triangle are at #(5 ,8 )# and #(9 ,1 )#. If the triangle's area is #36 #, what are the lengths of the triangle's sides?

Question

Two corners of an isosceles triangle are at #(5  ,8   )# and #(9   ,1   )#.  If the triangle's area is #36   #, what are the lengths of the triangle's sides?

Violet Aldrich · Accepted Answer

The length of three sides of triangle are #8.06 ,9.8 , 9.8# unit

Base of the isocelles triangle is #B= sqrt((x_2-x_1)^2+(y_2-y_1)^2)) = sqrt((9-5)^2+(1-8)^2)) =sqrt(16+49)=sqrt65 =8.06(2dp)#unit We know area of triangle is #A_t =1/2*B*H# Where #H# is altitude. #:. 36=1/2*8.06*H or H= 72/8.06=8.93(2dp)#unit Legs are #L = sqrt(H^2+(B/2)^2)= sqrt( 8.93^2+(8.06/2)^2)=9.80(2dp)#unit The length of three sides of triangle are #8.06 ,9.8 , 9.8# unit [Ans]

Jayden Jackson · Answer

undefined To find the lengths of the triangle's sides, we first need to determine the length of the base (the side that is not congruent) and then use trigonometry to find the lengths of the congruent sides (the legs).

Let's denote the points as follows: A(5, 8), B(9, 1), and C(x, y), where C is the vertex opposite the base.

Using the distance formula, we can find the length of side AB (the base):
AB = √((x2 - x1)² + (y2 - y1)²)
   = √((9 - 5)² + (1 - 8)²)
   = √((4)² + (-7)²)
   = √(16 + 49)
   = √65

Now, we can find the area of the triangle using the formula:
Area = 0.5 * base * height

Since the triangle is isosceles, the height will be the perpendicular distance from vertex C to side AB. We can use the distance formula again to find this distance.

Area = 0.5 * AB * height
36 = 0.5 * √65 * height
72 = √65 * height
height = 72 / √65

Now, we know the length of the base and the height. We can use trigonometry to find the lengths of the congruent sides (the legs). In a right triangle formed by the height, half of the base, and one of the legs, we have:
sin(θ) = opposite / hypotenuse

Where θ is the angle between the base and one of the legs. Since the triangle is isosceles, θ is also the angle formed by the base and one of the congruent sides.

We can find sin(θ) using the height and half of the base:
sin(θ) = height / (AB / 2)
sin(θ) = (72 / √65) / (AB / 2)
sin(θ) = (72 / √65) / (√65 / 2)

Now, we can solve for sin(θ) and then use it to find the lengths of the legs using the sine function.

sin(θ) = (72 / √65) / (√65 / 2)
sin(θ) = 2 * (72 / 65)
θ = sin⁻¹(2 * (72 / 65))

Once we have θ, we can find the lengths of the legs using the sine function again:
leg = height / sin(θ)

We repeat this process for both legs, using the same θ, since the triangle is isosceles.