Two corners of an isosceles triangle are at #(8 ,7 )# and #(2 ,3 )#. If the triangle's area is #64 #, what are the lengths of the triangle's sides?

Answer 1

See a solution process below:

The formula for the area of an isosceles triangle is:

#A = (bh_b)/2#

First, we must determine the length of the triangles base. We can do this by calculating the distance between the two points given in the problem. The formula for calculating the distance between two points is:

#d = sqrt((color(red)(x_2) - color(blue)(x_1))^2 + (color(red)(y_2) - color(blue)(y_1))^2)#

Substituting the values from the points in the problem gives:

#d = sqrt((color(red)(2) - color(blue)(8))^2 + (color(red)(3) - color(blue)(7))^2)#

#d = sqrt((-6)^2 + (-4)^2)#

#d = sqrt(36 + 16)#

#d = sqrt(52)#

#d = sqrt(4 xx 13)#

#d = sqrt(4)sqrt(13)#

#d = 2sqrt(13)#

The Base of the Triangle is: #2sqrt(13)#

We are given the area is #64#. We can substitute our calculation above for #b# and solve for #h_b#:

#64 = (2sqrt(13) xx h_b)/2#

#64 = sqrt(13)h_b#

#64/color(red)(sqrt(13)) = (sqrt(13)h_b)/color(red)(sqrt(13))#

#64/sqrt(13) = (color(red)(cancel(color(black)(sqrt(13))))h_b)/cancel(color(red)(sqrt(13)))#

#h_b = 64/sqrt(13)#

The Height of the Triangle is: #64/sqrt(13)#

To find the length of the triangles sides we need to remember the mid-line of an isosceles:
- bisects the base of the triangle into two equal parts
- forms a right angle with the base

Therefore, we can use the Pythagorean Theorem to find the length of the side of the triangle where the side is the hypotenuse and the height and #1/2# the base are the sides.

#c^2 = a^2 + b^2# becomes:

#c^2 = (1/2 xx 2sqrt(13))^2 + (64/sqrt(13))^2#

#c^2 = (sqrt(13))^2 + (64/sqrt(13))^2#

#c^2 = 13 + 4096/13#

#c^2 = 169/13 + 4096/13#

#c^2 = 4265/13#

#sqrt(c^2) = sqrt(4265/13)#

#c^2 = (sqrt(25)sqrt(185))/sqrt(13)#

#c^2 = (5sqrt(185))/sqrt(13)#

The Length of the Triangle's Side is: #(5sqrt(185))/sqrt(13)#

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Answer 2

To find the lengths of the sides of the isosceles triangle, we can first calculate the length of the base (the side opposite the vertex with equal angles). Then, we can use the given area to find the height of the triangle. Once we have the height, we can use the Pythagorean theorem to find the length of the equal sides.

  1. Calculate the length of the base:

    • Use the distance formula: (d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2})
    • Let the points be (A(8, 7)) and (B(2, 3)).
    • (d_{AB} = \sqrt{(8 - 2)^2 + (7 - 3)^2})
    • (d_{AB} = \sqrt{6^2 + 4^2})
    • (d_{AB} = \sqrt{36 + 16})
    • (d_{AB} = \sqrt{52})
  2. Use the area formula of a triangle: (Area = \frac{1}{2} \times base \times height)

    • Given (Area = 64) and base (d_{AB} = \sqrt{52}), we can solve for the height.
    • (64 = \frac{1}{2} \times \sqrt{52} \times height)
    • (height = \frac{64 \times 2}{\sqrt{52}})
    • (height = \frac{128}{\sqrt{52}})
  3. Now, for the length of the equal sides:

    • Using the Pythagorean theorem in one of the right triangles formed by the height, base, and one of the equal sides:
    • (s^2 = \left(\frac{1}{2} \times \sqrt{52}\right)^2 + \left(\frac{128}{\sqrt{52}}\right)^2)
    • Solve for (s), which represents the length of one of the equal sides.

This process yields the lengths of the sides of the isosceles triangle, where (s) is the length of each equal side, and the base is (\sqrt{52}).

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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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