Explanation:
3.
remember the law of sine :
a/sin(A) = b/sin(B) = c/sin(C)
a, b, c are the sides. A, B, C are the corresponding opposite angles.
and again, the sum of all angles in a triangle is always 180°.
so, the angle opposite of W = 180 - 90 - 60 = 30°.
and we get
6×sqrt(3)/sin(60) = W/sin(30) = W / 1/2 = 2W
W = 6/2 × sqrt(3)/sin(60) = 3×sqrt(3)/sin(60) =
= 3×sqrt(3) / sqrt(3)/2 = 6
X we get via Pythagoras
X² = (6×sqrt(3))² + W² = 36×3 + 6² = 108+36 = 144
X = sqrt(144) = 12
in the lower triangle the angle opposite of X = 180 - 90 - 45 = 45°
so, it is an isoceles triangle (both legs are equal).
therefore,
Y = X = 12
Z² = 12² + 12² = 144 + 144 = 288
Z = sqrt(288) = sqrt(16×18) = sqrt(16×9×2) =
= 4×3×sqrt(2) = 12×sqrt(2)
4.
x² + 2x - 1 = 2
x² + 2x + 1 = 4
(x + 1)² = 4
x + 1 = ±2
x = ±2 - 1
x1 = 2 - 1 = 1
x2 = -2 - 1 = -3
5.
in a rhombus all 4 sides are equal, and the opposite sides are parallel.
a.
so, PQ = QR
3x + 7 = -x + 17
4x = 10
x = 10/4 = 2.5
b.
the diagonals are perpendicular to each other (90°), intersect each other at their midpoints and cut their corner angles in half.
the angle RSM = angle PSM = 40°.
c.
the sum of all angles in any quadrilateral is always 360°.
in a rhombus the opposing angles are equal.
so,
the angle SPQ = angle SRQ
360 = SPQ + SRQ + PSR + PQR = 2×SPQ + 2×40 + 2×40 =
= 2×SPQ + 160
2×SPQ = 360 - 160 = 200
the angle SPQ = 200/2 = 100°.