Triangle ABC is inscribed in a circle that is inscribed in a square. If AB,AC and BC are 8,9 and 10 respectively, determine the exact area of the square?
The are of the square would be 110.82
Use Heron;s Rule:
The square will have sides 2R
Substitute values for a, b. c and solve
I hope my calculations are right
By signing up, you agree to our Terms of Service and Privacy Policy
To find the exact area of the square in which triangle ABC is inscribed, we'll first find the radius of the inscribed circle, then use it to find the diagonal of the square, and finally calculate the area of the square.

Radius of the inscribed circle (r): Use Heron's formula to find the area ((A)) of triangle ABC: [s = \frac{8 + 9 + 10}{2} = 13] [A = \sqrt{s(s  8)(s  9)(s  10)} = \sqrt{13(13  8)(13  9)(13  10)} = \sqrt{13 \times 5 \times 4 \times 3} = \sqrt{13 \times 60}] [A = \sqrt{780}]
Now, the area of a triangle is given by (A = \frac{1}{2} \times \text{base} \times \text{height}), but here the area is (A = \frac{1}{2} \times AB \times AC \times \sin(\angle BAC)). Since (\sin(\angle BAC) = \frac{BC}{2r}), we can rearrange it to solve for (r): [2r = \frac{BC}{\sin(\angle BAC)} = \frac{10}{\sin(\angle BAC)} = \frac{10}{\sin(\angle BAC)} = \frac{10}{\sqrt{780}} = \frac{10\sqrt{780}}{780}] [r = \frac{5\sqrt{780}}{780}]

Diagonal of the square (d): The diagonal of the square is twice the radius of the inscribed circle. So, (d = 2r): [d = 2 \times \frac{5\sqrt{780}}{780} = \frac{10\sqrt{780}}{780}]

Area of the square: The area of a square is given by (A_{\text{square}} = \frac{1}{2} \times d^2). Substitute the value of (d) into the formula: [A_{\text{square}} = \frac{1}{2} \times \left(\frac{10\sqrt{780}}{780}\right)^2 = \frac{1}{2} \times \frac{100 \times 780}{780^2} = \frac{100}{2 \times 780}] [A_{\text{square}} = \frac{50}{780} = \frac{25}{390}]
So, the exact area of the square in which triangle ABC is inscribed is (\frac{25}{390}) square units.
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a onesided 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 onesided 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 onesided 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 onesided 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.
 A circle has a chord that goes from #( pi)/3 # to #(7 pi) / 12 # radians on the circle. If the area of the circle is #36 pi #, what is the length of the chord?
 A triangle has corners at #(9 ,5 )#, #(2 ,5 )#, and #(3 ,6 )#. What is the area of the triangle's circumscribed circle?
 A circle has a center that falls on the line #y = 3/8x +5 # and passes through # ( 7 ,3 )# and #(2 ,9 )#. What is the equation of the circle?
 What is the equation of the circle with a center at #(5 ,3 )# and a radius of #2 #?
 A circle has a chord that goes from #( 5 pi)/3 # to #(17 pi) / 12 # radians on the circle. If the area of the circle is #27 pi #, what is the length of the chord?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7