Two corners of an isosceles triangle are at #(8 ,5 )# and #(6 ,2 )#. If the triangle's area is #4 #, what are the lengths of the triangle's sides?

Answer 1

Lengths of triangle's sides are # 3.61(2dp) , 2.86(dp), 2.86(dp)# unit.

Length of base of isoceles triangle is #b=sqrt((x_1-x_2)^2+(y_1-y_2)^2) = sqrt((8-6)^2+(5-2)^2)= sqrt(4+9)=sqrt 13=3.61(2dp)#
Area of isoceles triangle is # A_t = 1/2 *b*h or 4 = 1/2*sqrt13*h or h = 8/sqrt 13 =2.22(2dp)#. Where #h# be the altitude of triangle.
Legs of isoceles triangle are #l_1=l_2= sqrt( h^2 +(b/2)^2) =sqrt( 2.22^2 +(3.61/2)^2) = 2.86(2dp)#unit
Lengths of triangle's sides are # 3.61(2dp) , 2.86(dp),2.86(dp)# unit. [Ans]
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Answer 2

To find the lengths of the sides of the isosceles triangle, we can first calculate the distance between the two given points, which will give us the base of the triangle. Then, we can use the formula for the area of a triangle to find the height, and thus, the lengths of the remaining sides.

  1. Calculate the distance between the two given points: [ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ] [ \text{Distance} = \sqrt{(6 - 8)^2 + (2 - 5)^2} ] [ \text{Distance} = \sqrt{(-2)^2 + (-3)^2} ] [ \text{Distance} = \sqrt{4 + 9} ] [ \text{Distance} = \sqrt{13} ]

  2. The area of a triangle is given by the formula: [ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} ] [ 4 = \frac{1}{2} \times \sqrt{13} \times \text{height} ]

Solving for the height: [ 8 = \sqrt{13} \times \text{height} ] [ \text{height} = \frac{8}{\sqrt{13}} ]

Now, we can use the Pythagorean theorem to find the lengths of the other sides of the triangle since it's an isosceles triangle.

Let ( b ) be the length of one of the congruent sides: [ b^2 = \left(\frac{\sqrt{13}}{2}\right)^2 + \left(\frac{8}{\sqrt{13}}\right)^2 ]

Solving for ( b ): [ b^2 = \frac{13}{4} + \frac{64}{13} ] [ b^2 = \frac{169}{4} ] [ b = \frac{\sqrt{169}}{2} ] [ b = \frac{13}{2} ]

Therefore, the lengths of the sides of the triangle are ( \frac{13}{2} ), ( \frac{13}{2} ), and ( \sqrt{13} ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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