Straight Lines: questions with answers

The formula $PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ is Pythagoras on the right triangle whose legs are the two separations. 25 is the SQUARE of the distance; the root is what turns it back into a length, here 5 units.

Parallel is the simple condition $m_1 = m_2$, so the answer is the SAME slope, 3/4. The negative reciprocal -4/3 belongs to perpendicularity: it would turn the line through a right angle instead of laying a second one beside it.

Turning a line through a right angle inverts the slope AND changes its sign: $m_2 = -\frac{1}{m_1}$. Here $m_1 =$3/4, so $m_2 =$-4/3 and the product is −1. Taking the reciprocal alone gives 4/3, whose product with 3/4 is +1 -- that is the reflection of PQ in the line y = x, not a perpendicular to it.

Of the three marked angles only the one opening anticlockwise from the POSITIVE x-axis is the inclination, here 150°. The angle at the y-axis, 60°, is 90° away from it, and 30° is its supplement measured on the other side.

The modulus is what picks the ACUTE angle out of the supplementary pair. The obtuse angle φ is 180° − θ, and its tangent is the negative, -6/7 -- which is what the formula returns when the modulus is dropped.

A line parallel to the x-axis holds y fixed. Reading the OTHER coordinate off the point gives y = -4, a line through a different place; naming the other LETTER gives x = -2, a line at right angles to this one.

Reading a slope off a drawing is the same division as reading it off coordinates: the vertical change goes on top. Counting the squares ACROSS first and dividing by the squares UP gives -2/3, the reciprocal, which is the commonest way to lose this.

Each pair meets, but the three pairwise intersections are different. The first two lines meet at (-2, 1), which does not satisfy the third equation.

The form subtracts the COORDINATE, and subtracting -6 adds 6. Reading x − x₀ as 'x minus the number shown' gives the line through (6, 5) instead -- and substituting P into it is the check that catches the slip at once.

Two distinct points determine one and only one line. The two ways of labelling them give the SAME slope 5/3, not two different lines -- which is why the two-point form has one answer however it is written.

The number under $x$ is the X-intercept, 2, and the number under $y$ is the Y-intercept, -4. Exchanging them gives x/(-4) + y/2 = 1, a line through (-4, 0) and (0, 2) instead.

The constant term is the Y-intercept: it is what remains when $x=0$. B, the crossing on the x-axis, is at -1, and the two are equal only in the trivial case where both are 0.

The line through P at 135° has slope −1, so it satisfies $x+y=14$. Its intersection with the given line is Q(14/3, 28/3). The absolute changes in x and y are both 16/3, so PQ has length $\frac{16\sqrt{2}}{3}$. The perpendicular distance answers a different question.

The shortest separation between parallel lines is perpendicular to both. Segment Q is perpendicular; the vertical segment P is longer because the lines are not horizontal.

The denominator is a PYTHAGOREAN combination of the two coefficients, √(A²+B²) = 5. Adding them instead gives (-3) + (-4), and leaving the root off gives 25; both scale the answer by a factor that has nothing to do with the geometry.