Triangle ABC has ∠A=40∘, ∠B=60∘, ∠C=80∘. Points M,N trisect the side BC and points P,Q trisect the side AC. The lines AM,AN,BP,BQ intersect at the points S,T,U,V as shown in the figure below, dividing the triangle into 9 regions. Determine the smallest possible value of [ABC]+[STUV] such that both [ABC] and [STUV] are positive integers.
plz answer it, i need to know by tonight
35
Draw a triangle 8 age what is draw line parallel to the opposite side
To determine the value of [ABC]+[STUV], we need to find the areas of triangle ABC and STUV separately.
Let's start with triangle ABC. We have the angles ∠A = 40∘, ∠B = 60∘, and ∠C = 80∘. Notice that the sum of the angles in triangle ABC is 180∘, which means it's a valid triangle.
To find the area of triangle ABC, we can use the formula: [ABC] = 1/2 * a * b * sin(C), where a and b are the lengths of two sides of the triangle, and C is the included angle between them.
Given that triangle ABC is not necessarily an isosceles triangle, we need to find the lengths of sides AB and AC.
Let's assume that side AB has length x. Since angle ∠A = 40∘ and angle ∠B = 60∘, we can use the law of sines to find the length of side AC.
sin(A)/a = sin(B)/b = sin(C)/c
sin(40∘)/x = sin(60∘)/AC
Using the values of sin(40∘) and sin(60∘) from a trigonometric table or using a calculator, we can solve for AC.
Now that we have the lengths of sides AB and AC, we can calculate the area [ABC].
Moving on to triangle STUV, we can divide it into three smaller triangles: AST, MSV, and TUV.
To find the area of each of these triangles, we can use the formula: [Triangle] = 1/2 * base * height.
For triangle AST, the base is ST and the height can be obtained by dropping a perpendicular from point S to side AT.
For triangle MSV, the base is SV and the height can be obtained by dropping a perpendicular from point M to side VU.
For triangle TUV, the base is TU and the height can be obtained by dropping a perpendicular from point V to side UQ.
Using the lengths of sides and the heights, we can calculate the areas of these smaller triangles.
Finally, the area of [STUV] can be obtained by summing up the areas of these three smaller triangles.
Once we have calculated the areas of [ABC] and [STUV], we can determine the smallest possible value of [ABC]+[STUV] such that both [ABC] and [STUV] are positive integers.