Explanation:
always remember : the sum of all angles in every triangle is 180°.
3a)
the small right-angled triangle to the left has the same third angle (top left corner) as the large triangle.
for this small triangle we get
180 = 90 + 40 + angle
angle = 50°
note we can use the law of sine :
sin(A)/a = sin(B)/b = sin(C)/c
a, b, c are the sides, A, B, C are the corresponding opposite angles.
therefore,
sin(theta)/15 = sin(50)/22
sin(theta) = 15×sin(50)/22 = 0.522303029...
theta = 31.48686133...° ≈ 31°
3b)
the supplementary angle of 52° (together they have 180°, as they cover together the whole range of angles around a single point on one side of a line) is
180 - 52 = 128°
this is the large angle in the 11cm triangle.
the third angle in the 52° right-angled triangle is given by
180 = 90 + 52 + angle
angle = 38°
the law of sine gives us the other 2 sides of that triangle :
sin(52)/9 = sin(38)/bottom line = sin(90)/baseline
bottom line = 9×sin(38)/sin(52) = 7.031570639... cm
baseline = 9/sin(52) = 11.42116394... cm
so, the bottom line of the large triangle is
11 + 7.031570639... = 18.031570639... cm
the baseline of the large triangle is then
baseline² = 9² + 18.031570639...² = 406.1375397...
baseline = 20.15285438... cm
now,
sin(supplement of theta)/9 = sin(90)/20.15285438...
sin(supplement of theta) = 9/20.15285438... =
= 0.446586862...
supplement of theta = 26.52491052...°
theta = 180 - supplement of theta = 153.4750895...° ≈
≈ 153°
7)
there are 4 basic transformations :
rotation, translation, reflection, dilation
f(x) = x²
is a parabola with vertex at (0, 0) and opening up.
a rotation changes the direction of the opening of the parabola. depending on the center of the rotation (different than the origin) it would also move the vertex away from the original position.
a translation just shifts the parabola up, down, left or right without changing any of its form or orientation.
a reflection used a line (like a coordinate axis) as mirror and mirrors the graph to the other side of the line.
a left/right reflection just moves the graph, since the parabola is symmetric, and so, and left/right flip does not change the form of the curve.
an up/ down reflection reverses the orientation (the opening direction) of the parabola : up to down and vice versa.
a dilation changes the size, the distance between the points of the curve, making it larger or smaller, but is keeping the orientation. depending on the center of dilation (projection) the resulting curve might also move the curve.