Area Of A Triangle
Author: Edward M. Knote
Date: 11/10/06

In this worksheet you will investigate how to find the area of a triangle.

Investigation:
1)  Move point D so that the triangle is a right triangle, then move the Hint slider, describe the end result. 
2)  Move point D to make an obtuse triangle and move the Hint slider, describe the end result.  (perform this task for an acute triangle as well)
3)  What conjecture can you make about the formula for area of a triangle?  Test this conjecture with 5 different base and height combinations and record your results.

bok

wysokość

wysokość

obrót

P = \frac{\textcolor{204,0,0}{a}\cdot \textcolor{204,0,0}{ h}}{2}

wzór

a

h

Pole trójkąta

A = (-6.383333333333333, 3.1333333333333346)
wysokość = 10 (slider from 5 to 10, step 1)
bok = 15 (slider from 10 to 15, step 1)
c = Circle(A, wysokość)
d = Circle(A, bok)
B = Point(d)
e = Segment(A, B)
f = Line(A, B)
g = OrthogonalLine(A, f)
C = Intersect(c, g, 1)
h, a = Tangent(C, c)
D = Point(h)
P, d_1, a_1, b_1 = Polygon(A, B, D)
i = OrthogonalLine(D, f)
E = Intersect(f, i)
j = Segment(E, D)
k = Segment(E, B)
F = Midpoint(D, B)
G = Midpoint(D, A)
H = Point(a_1)
p = Circle(B, H)
I = Intersect(p, d_1, 1)
l = OrthogonalLine(H, a_1)
m = OrthogonalLine(I, d_1)
J = Intersect(l, m)
K = If(x(D) ≟ x(B), G)
L = If(x(D) ≟ x(B), J)
n = Segment(L, H)
q = Segment(L, I)
α = 0 rad (slider from 0*° to 3.141592653589793, step 0.017453292519943295)
A' = Rotate(A, α, K)
B' = Rotate(B, α, K)
D' = Rotate(D, α, K)
P', d', a', b' = Polygon(A', B', D')
M = If(x(D) ≟ x(A), F)
N = Point(b_1)
r = Circle(A, N)
O = Intersect(r, d_1, 1)
s = OrthogonalLine(N, b_1)
t = OrthogonalLine(O, d_1)
Q = Intersect(s, t)
R = If(x(D) ≟ x(A), Q)
c_1 = Segment(N, R)
e_1 = Segment(R, O)
A'_1 = Rotate(A, -α, M)
B'_1 = Rotate(B, -α, M)
D'_1 = Rotate(D, -α, M)
P'_1, d'_1, a'_1, b'_1 = Polygon(A'_1, B'_1, D'_1)
S = If(x(D) > x(B), G)
A'_2 = Rotate(A, α, S)
B'_2 = Rotate(B, α, S)
D'_2 = Rotate(D, α, S)
P'_2, d'_2, a'_2, b'_2 = Polygon(A'_2, B'_2, D'_2)
T = If(x(D) < x(A), F)
A'_3 = Rotate(A, -α, T)
B'_3 = Rotate(B, -α, T)
D'_3 = Rotate(D, -α, T)
P'_3, d'_3, a'_3, b'_3 = Polygon(A'_3, B'_3, D'_3)
U = If((x(D) > x(A)) ∧ (x(D) < x(B)), F)
A'_4 = Rotate(A, -α, U)
B'_4 = Rotate(B, -α, U)
D'_4 = Rotate(D, -α, U)
P'_4, d'_4, a'_4, b'_4 = Polygon(A'_4, B'_4, D'_4)
V = Midpoint(D, E)
W = Midpoint(A, B)
b = false
Z = Intersect(d_1, j)
A_1 = Intersect(d_1, b_1)
B_1 = Intersect(a_1, a'_4)
C_1 = Intersect(d_1, a_1)
D_1 = Midpoint(A_1, C_1)
E_1 = Midpoint(D, Z)