An isosceles triangle has sides A, B, and C with sides B and C being equal in length. If side A goes from #(7 ,1 )# to #(8 ,5 )# and the triangle's area is #32 #, what are the possible coordinates of the triangle's third corner?
The possible points are
Given:
The endpoints of side A are Let side Find the length of the base: The area of a triangle is: Substitute in the values of Area and Base: Compute midpoint of side A Find the two points that are the same distance as height away from the midpoint: We need the slope of side A: The slope of the height is perpendicular to side A: Use the point slope form of the equation of a line to write the equation for the height: With equations [1] and [2] this ugly, I am going to use WolframAlpha to solve them: The possible points are Here is an image of all of the parts of this problem that should show you that the answer is correct:
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We solve this Problem deploying the methods of Analytic Geometry.
Recall that, the Area of a Triangle having vertices
Enjoy Maths.!
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Since the triangle is isosceles, let's denote the third corner as ( D ). ( BD ) and ( CD ) are equal sides.
The length of side ( A ) can be found using the distance formula between two points.
[ A = \sqrt{(8 - 7)^2 + (5 - 1)^2} = \sqrt{1^2 + 4^2} = \sqrt{17} ]
The area of the triangle can be found using the formula:
[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} ]
Given that the area is 32 and the base is ( \sqrt{17} ), we can solve for the height:
[ 32 = \frac{1}{2} \times \sqrt{17} \times \text{height} ]
[ \text{height} = \frac{64}{\sqrt{17}} ]
Now, the coordinates of point ( D ) can be found by moving up or down from point ( (8, 5) ) by a distance equal to ( \frac{64}{\sqrt{17}} ), keeping in mind that the triangle is isosceles, and hence the points ( B ) and ( C ) lie on a line perpendicular to ( AD ).
So, the possible coordinates of the third corner ( D ) are ( (8, 5 + \frac{64}{\sqrt{17}}) ) and ( (8, 5 - \frac{64}{\sqrt{17}}) ).
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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.
- Circle A has a center at #(1 ,8 )# and an area of #32 pi#. Circle B has a center at #(2 ,3 )# and an area of #28 pi#. Do the circles overlap?
- A line passes through #(5 ,9 )# and #(7 ,3 )#. A second line passes through #(3 ,1 )#. What is one other point that the second line may pass through if it is parallel to the first line?
- What is the perimeter of a triangle with corners at #(1 ,2 )#, #(8 ,3 )#, and #(4 ,1 )#?
- Circle A has a center at #(-5 ,7 )# and a radius of #4 #. Circle B has a center at #(7 ,2 )# and a radius of #3 #. Do the circles overlap? If not, what is the smallest distance between them?
- What is the perimeter of a triangle with corners at #(6 ,5 )#, #(9 ,1 )#, and #(3 ,8 )#?

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