Find the area not grazed in this triangular plot.?

Question

A farmer had a triangular plot measuring 15m, 20m, 25m.the sides of the triangle. He kept his three horses attached to three poles at the vertices of this plot by a rope 7 m. How much of the plot is ungrazed (grass not eaten)?
If you answer this question, you qualify for IIT. I request all IIT students not to answer, please.

Answer 1

Area of the triangle - area of sectors.

Since the sides of the triangle are 15, 20, and 25, this is a
3-4-5 right triangle. The hypotenuse is 25 m, the longest side.

The area of the triangle is 1/2 base * height

At = 1/2 15m 20m
At = 150 m^2

Since the 3 angles are the vertices of a triangle, their measures add up to 180 degrees, or half a circle. The radius of 7 m is less than half the shortest side, so none of the grazed areas overlap.

The area of the half circle is half the area of the circle with radius 7 m.

Ahc = 1/2 pi (7m)^2
Ahc = 1/2 pi 49 m^2
Ahc = 49/2 pi m^2

A = Area of the triangle - area of sectors
A = 150 m^2 - 49/2 pi m^2
A = (150 - 49/2 pi) m^2

Answer 2

This is a right traingle (3,4,5 - times 5). None of the grazed areas overlap since 7 is less than half of any of the lengths of the sides. The ungrazed area is then the area of the triangle minus the sum of three circular arc areas.

The circular arc areas can simply be added by noting that they sum to an angle of 180 degrees (they are a triangle) and that is pi radians. Since there are 2pi radians in a circle, the area of the arcs is 1/2 the area of a circle of radius 7m.

Area = (1/2)(pi*7^2) = 24.5pi m^2

Area of the triangle is (1/2)base*height = (1/2)(15)(20) = 150m^2

Area not grazed = 150 - 24.5pi = 73.03 m^2

Answer 3

Ok, we know that none of the grazing areas will overlap because, 2 * 7 = 14 and none of the lengths are that small. So, this also makes it a little easier on us.

Now we find the area of a circle with a 7m radius

pi(7)squrd = 49pi

Since we know that all the angles of a triangle add up to 180 we need to use this to our advantage.

a circle has 360 degrees and a triangle has 180 so if we were to add up all the angles of the triangle that were grazed we'd get 180.

so let's make a ratio and multiply it by the circles area.

(180/360) (49pi) = .5(49pi)

= 24.5pi or 76.97squr m

Now, we know that a 15/20/25 triangle is a right triangle because of the 345 rule. So, we just need to find the area and subtract 76.97 for our answer.

(15 *20)/2 = 150 squr m

150 - 76.97 = 73.03 squr m of ungrazed grass

:)

Answer 4

The required area = Area of Triangle – Total Area of arcs made at the vertices.
Sides are 15, 20 and 25 cm.
Here 25^2 = 15^ +20^2
625 = 225 + 400 = 600 This is a right angled triangle.
Hypotenuse 25 and sides 15 and 20 cm
Area of triangle = 1/2 Base × Height
= 1/2 × 15 × 20 = 150 sq,cm.
To calculate the area of arcs.
One of the angle is 90 Let other angles be x and y
Area of arc made at vertex having 90 degree = 90/360 pi7^2
Area of arc made at vertex having x degree = x/360 pi7^2
Area of arc made at vertex having y degree = y/360 pi7^2
Therefore total area = sum of area of three arcs.
= (90 + x + y)/360 × pi7^2 = (90 + x + y)/360 × 154 (pir^2 = 154 (Value of pi = 22/7)
But (90 + x + y) = 180 degree i.e. sum of three angles of a triangle.
Then Area of arc = 154 × 180/360 = 154/2 = 77 cm^2
So the ungraze part = 150 – 77 = 73 cm^2