In the right triangle shown,
m
∠
Q
=
60
°
m∠Q=60°m, angle, Q, equals, 60, degree and
Q
R
=
2
3
QR=2
3

Q, R, equals, 2, square root of, 3, end square root.

A) 2
B) 4
C) 6
D) 4 square root of 3
E) 12

Step-by-step explanation:

Vertices

(c, 0) | ((-a^2 + b^2 + c^2)/(2 c), sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(2 c)) | (0, 0)

Number of vertices

3

Variable constraints

a>0 and b>0 and c>0 and a + b>c and b + c>a and a + c>b

Diagonal lengths

(data not available)

Height

sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(2 c)

Area

A = 1/4 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))

Centroid

x^_ = ((-a^2 + b^2 + 3 c^2)/(6 c), sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(6 c))

Mechanical properties:

Area moment of inertia about the x-axis

J_x invisible comma x = (-(a - b - c) (a + b - c) (a - b + c) (a + b + c))^(3/2)/(96 c^2)

Area moment of inertia about the y-axis

J_y invisible comma y = (sqrt(-(a - b - c) (a + b - c) (a - b + c) (a + b + c)) (4 c^2 (b^2 - a^2) + (a^2 - b^2)^2 + 7 c^4))/(96 c^2)

Polar moment of inertia

J_zz = -1/48 sqrt(-(a - b - c) (a + b - c) (a - b + c) (a + b + c)) (a^2 - 3 (b^2 + c^2))

Product moment of inertia

J_x invisible comma y = -((a - b - c) (a + b - c) (a - b + c) (a + b + c) (a^2 - b^2 - 2 c^2))/(96 c^2)

Radii of gyration about coordinate axes

r_x = ((a + b - c) (a - b + c) (-a + b + c) (a + b + c))^(1/4)/(sqrt(6) c)

r_y = sqrt(4 c^2 (b^2 - a^2) + (a^2 - b^2)^2 + 7 c^4)/(sqrt(6) c ((a + b - c) (a - b + c) (-a + b + c) (a + b + c))^(1/4))

Distance properties:

Side lengths

a | b | c

Perimeter

p = a + b + c

Inradius

r = 1/2 sqrt(-((a - b - c) (a + b - c) (a - b + c))/(a + b + c))

Circumradius

R = (a b c)/sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))

Generalized diameter

max(a, b, c)

Convexity coefficient

χ = 1

Mean triangle area

A^_ = 1/48 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))

mathpapa

step-by-step explanation: