0
r = ( BC2 - BC2/AB )½ / (1+√2)
Proof">Proof">ProofIt is easy to see that the radius of the inscribed circle must touch the edges of the triangle at the hypotenuse, height, and base. Furthermore, radius lines that extend to the base (AC) and to the height (BC) of the triangle must be at right angles to one another. Finally, the radius that extends to the hypotenuse (AC) if continued outward will reach point C of the triangle, dividing the angle in half.Let the point at which the radius of the inscribed circle touches the hypotenuse (AC) of the triangle be called Z.
This divides the original triangle into two similar triangles, CBZ and ACZ.
Applying the principle of similar triangles, we know that:
(1) BC / AB = ZB / BC solving for ZB,
(2) ZB = BC2 / AB
The length of the line ZC can be found using the Pythagorean theorem:
(3) ZC = √( BC2 - ZB2 )
However, since the radius of the inscribed circle passes through ZC we can write ZC in terms of that radius. The formula below becomes more evident when you draw a picture to illustrate the problem.
(4) ZC = r ( 1 + √2 )
We can now combine (4) and (3) to solve for r.
(5) √( BC2 - ZB2 ) = r ( 1 + √2 ) solving for r yields:
(6) r = √( BC2 - ZB2 ) / (1+√2) substituting (2) for ZB yields the final formula
(7) r = ( BC2 - BC2/AB )½ / (1+√2)
The answer makes the following statement:
Finally, the radius that extends to the hypotenuse (AC) if continued outward will reach point C of the triangle, dividing the angle in half.
This is not necessarily a true statement. It is only true for a 45 degree, 45 degree, 90 degree triangle. The formula given does not give the correct answer of 2 for a 6, 8, 10 triangle.
For a formula that works on all rt. triangles:
if c is the hypotenuse and a and b are the other 2 sides
ab/a+b+c or
a+b-c/2
you should get the same answer
What else can I help you with?
Which is true about a circle is inscribed in a hexagon?
The radius of a circle inscribed in a regular hexagon equals the length of one side of the hexagon.
What is the area of a square inscribed in a circle with the radius 6?
The answer is 72, i think!
An equilateral triangle surrounds a circle of radius 10 cm and is itself surrounded by a larger circle Find the radius of the bigger circle?
Make a sketch of the situation. From a corner of the equilateral triangle draw a radius of the large circle, and from an adjacent side draw a radius of the smaller circle. You should have formed a small right-angled triangle with a known side of 10cm. and known angles of 30o, 60o and 90o. (The interior angles of an equilateral triangle are each 60o.) The hypotenuse is the unknown radius of the larger circle. But since cos 60 = 0.5, it is evident that the hypotenuse is 20cm. long.
A chord of a circle is 9 inches long and its midpoint is 6inches from the center of the circle What is the radius of the circle?
Half of the chord, the distance of the midpoint from the center, and a radius, form a right triangle, with the radius as its hypotenuse. (4.5)2 + (6)2 = (radius)2 (20.25) + (36) = 56.25 = R2 Radius = 7.5 inches
A chord of a circle has length 4.2 cm and is 8 cm from the center of the circle what is the radius of the circle?
To find the radius of the circle, we can use the Pythagorean theorem. The chord divides the circle into two equal parts, each forming a right triangle with the radius. The radius, the distance from the center to the chord, and half the length of the chord form a right triangle. Using the Pythagorean theorem, we have (radius)^2 = (distance from center)^2 + (1/2 * chord length)^2. Substituting in the given values, we get (radius)^2 = 8^2 - (1/2 * 4.2)^2. Solving for the radius gives us a radius of approximately 7.48 cm.
The shortest distance from the center of the inscribed circle to the triangle's sides is the circle's?
radius
Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius a in C programming?
Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius a in C programming
Does a triangle have a radius?
It has at least two radii, the radius of the circle going through the vertices and the radius of the inscribed circle touching all the sides.
The shortest distance from the center of the circumscribed circle to the vertices of the inscribed triangle is the circle's radius?
True
What is the formula for equillateral triangle?
There are different formula for: Height, Area, Perimeter, Angle, Length of Median Radius of inscribed circle Perimeter of inscribed circle Area of inscribed circle etc.
The shortest distance from the center of the circumscribed circle to the sides of the inscribed triangle is the circles radius?
FALSE
Is the shortest distance from the center of the circumscribed circle to the vertices of the inscribed triangle is the circles radius?
True
Is the distance from the point of concurrency of the angle bisectors of a triangle to a point on the inscribed circle is the radius of the circle True or False?
false
How do you draw an inscribed circle using a compass?
To draw an inscribed circle (incircle) of a triangle using a compass, first, construct the triangle and find the angle bisectors of at least two angles. Where these bisectors intersect is the incenter, which is the center of the inscribed circle. Set the compass point on the incenter, adjust the radius to reach one of the triangle's sides, and draw the circle. This circle will touch all three sides of the triangle at their respective points, completing the inscribed circle.
Is the distance from the point of concurrency of the angle bisectors of a triangle to a point on the inscribed circle is the radius of the cirlce?
yes
What is the area of the rectangle of largest area that can be inscribed in a circle of radius r?
It is 2*r^2.
The radius of a incircle is 14cm what is the side of the equilateral triangle abc?
Where the side of the equilateral triangle is s and the radius of the inscribed circle is r:s = 2r * tan 30° = 48.50 cm
