As we can see, AC and BC are negative reciprocals, which means they are perpendicular segments.
4/3(- 3/4)=- 1
Therefore, this is in fact a right triangle.
By calculating the segment lengths we can also decide if the triangle is isosceles. We can do this by substituting the segments' endpoints into the Distance Formula.
Segment
Points
sqrt((x_2-x_1)^2+(y_2-y_1)^2)
d
AB
A(3,0), B(2,7)
sqrt(( 3- 2)^2+( 0- 7)^2)
sqrt(50)
AC
A(3,0), C(6,4)
sqrt(( 3- 6)^2+( 0- 4)^2)
5
BC
B(2,7), C(6,4)
sqrt(( 2- 6)^2+( 7- 4)^2)
5
Since BC and AC have the same length, this is an isosceles triangle.
b From Part A, we know that this is a right isosceles triangle. According to the Base Angles Theorem, it has two congruent base angles. By using the Triangle Angle Sum Theorem we can write the following equation.
m∠ A+m∠ B+90^(∘)=180^(∘)
Since m∠ B=m∠ A, we can solve this equation.
c From Part A we know that △ ABC is an isosceles right triangle. From Part B we know that m ∠ A = 45 ^(∘). Let's recall the definition of tangent of an acute angle in a right triangle.
tan( θ ) = Opposite/Adjacent
Let's find which side is opposite and which is adjacent to angle ∠ A.
We see that BC is opposite and AC is adjacent to angle ∠ A. Since m∠ A = 45 ^(∘) and from Part A we also know that BC = 5 and AC = 5, we can find the value of tan( 45 ^(∘)).
