Slope triangles: why a line has one slope everywhere
You measured how steep a line is between two of its points. Why would any other two points give the same number?
Where you are 4 in Slope
Why this exists
The Slope of a line is its rise divided by its run between two of its points. But which two points? If two points on the left gave one number and two on the right gave another, there would be no such thing as “the slope of the line”, only “the slope between these two points”.
In Slope from a graph or a table and The slope formula and the four kinds of slope the same number came out whichever pair was chosen. The reason: every pair of points on a line makes a right triangle with the line, and all those triangles are scaled copies of one another, that is, similar. Similar triangles keep their sides in the same ratio, and rise over run is such a ratio.
Definition
Take a line that is neither vertical nor horizontal, and two points on it. Draw a horizontal segment from one point and a vertical segment from the other, so that they meet. The three segments make a right triangle whose longest side, the hypotenuse, lies along the line. This is a slope triangle of the line.
The horizontal leg is the run and the vertical leg is the rise. In this lesson both are lengths, so both are positive numbers; the direction is read off separately. The slope of the line is
when the line goes up from left to right, and the same fraction with a minus sign in front when the line goes down from left to right.
The same two points give two slope triangles, one on each side of the line, depending on which point the horizontal leg starts from. Both count, and the Results below cover both.
Intuition
In Figure 1 the small triangle has run 3 and rise 2, and the large one has run 9 and rise 6. Every side of the large one is three times the matching side of the small one: it is a scale drawing of it with scale factor 3. Both give
Grow or shrink a slope triangle from its corner on the line and the line stays where it is, so the hypotenuse stays on it. Slide the triangle along the line and it still fits. Neither move changes the shape, so the ratio of the legs cannot change.
Results
Slope triangles are similar. Any two slope triangles of the same line are similar. If one has run a and rise b and the other has run c and rise d, then
So the slope of a line is the same whichever two points of the line are used to find it.
The scale factor. If one slope triangle’s run is k times another’s, its rise is also k times the other’s rise, and so is its hypotenuse.
A test for a bend. If points P, Q and R are taken in order, and the slope from P to Q is not the slope from Q to R, then the three points do not lie on one line.
Why it is true
Let \ell be a line that rises to the right. Draw two slope triangles under it, \triangle ABC and \triangle DEF, with A, B, D, E on the line and the right angles at C and F, as in Figure 2.
The right angles match. One leg of each triangle is horizontal and the other is vertical, and a horizontal and a vertical segment meet at a right angle. So \angle C = \angle F = 90^\circ.
The slanted angles match. The legs AC and DF are both horizontal, so the lines they lie on are parallel. The line \ell crosses both of them: it is a transversal. The angle \angle CAB sits at the lower crossing, between the horizontal pointing right and \ell pointing up, and \angle FDE sits in the same position at the upper crossing. They are corresponding angles, and corresponding angles on parallel lines cut by a transversal are equal. So \angle CAB = \angle FDE.
Two pairs of equal angles are enough. By angle-angle similarity,
with A matching D, B matching E and C matching F.
The ratio follows. Similar triangles have one scale factor k for every pair of matching sides, so DF = k \cdot AC and EF = k \cdot BC. Then
and the factor k cancels. Rise over run is the same for both triangles.
A triangle on the other side. Suppose the slope triangle on AB is drawn above the line instead, with its corner G directly above A and level with B. Then A, C, B and G are the corners of a rectangle, since all four of its angles are right angles, and opposite sides of a rectangle are equal: AG = CB and GB = AC. The upper triangle has exactly the same rise and run as the lower one. So any triangle above the line can be swapped for its partner below the line, and the argument above applies to the partner.
A line that falls. If \ell falls to the right, reflect the whole picture in a vertical line. A reflection keeps every length, keeps horizontal segments horizontal and vertical ones vertical, and turns \ell into a line that rises to the right. The argument above applies there, and the lengths it compares are the original ones.
The sign. Every horizontal line meets \ell at the same angle, opening the same way, as the corresponding angles showed. So a line that goes up to the right at one place goes up to the right everywhere, and the sign of the slope does not change either.
The test for a bend. If P, Q and R were on one line, the triangles on PQ and QR would both be slope triangles of it and give the same ratio. So different ratios mean a bend.
Worked examples
Checking two triangles on a grid
In Figure 1, show that the two slope triangles are similar, give the scale factor, and find the slope of the line.
SolutionBoth triangles are slope triangles of one line, so they are similar. The small one has run 3 and rise 2; the large one has run 9 and rise 6. Matching legs give the scale factor:
Both legs are multiplied by 3, and so is the hypotenuse. The slope is the same from either triangle:
The line goes up to the right, so the slope is positive.
A missing leg
A line goes up to the right. One of its slope triangles has run 7 and rise 4. Another slope triangle of the same line has run 17.5. Find its rise. A third has rise 3; find its run.
SolutionAll slope triangles of one line are similar, so they share the ratio of rise to run. Call the missing rise h:
Multiplying both sides by 17.5 gives \displaystyle h = \frac{70}{7} = 10. Seen as a scale factor: the run went from 7 to 17.5, which is multiplying by 2.5, so the rise goes from 4 to 4 \times 2.5 = 10.
For the third triangle call the run r:
The rise went from 4 to 3, a scale factor of \displaystyle \frac{3}{4} = 0.75, less than 1 because this triangle is smaller. So the run goes from 7 to 7 \times 0.75 = 5.25.
Check. \displaystyle \frac{10}{17.5} = \frac{100}{175} = \frac{4}{7} and \displaystyle \frac{3}{5.25} = \frac{300}{525} = \frac{4}{7}.
Triangles on opposite sides
In Figure 3 a line falls to the right through the grid corners P, Q, R and S. One slope triangle sits below the line on PQ and a larger one sits above the line on RS. Show that they are similar and find the slope.
The small triangle has a vertical leg of 1 and a horizontal leg of 3. The large triangle has a horizontal leg of 9 and a vertical leg of 3.
The large triangle is on the other side of the line, so first swap it for its partner. The triangle below the line on RS, going down 3 from R and then across 9 to S, makes a 9 by 3 rectangle with the one above, so it has the same legs. That partner and the small triangle are both slope triangles below the same line, so they are similar, and so the large triangle is similar to the small one too. The scale factor is
The ratio of vertical to horizontal leg is \displaystyle \frac{1}{3} for the small triangle and \displaystyle \frac{3}{9} = \frac{1}{3} for the large one. The line falls from left to right, so
A roof
The straight top edge of a roof rises 5 for every 12 it goes across: a builder calls this a “5 in 12” pitch. Svanhild measures from the eave, the low edge of the roof. At 3 m in from the eave, measured horizontally, the roof edge is 1.25 m above the eave. (a) The ridge is 7.2 m in from the eave. How far is it above the eave? (b) A vent has to go where the roof edge is 2 m above the eave. How far in is that?
SolutionThe roof edge is a straight line, and every triangle drawn from the eave across and up to a point on the edge is a slope triangle of it, similar to the builder’s 12 by 5 one. So each has rise over run \displaystyle \frac{5}{12}. Svanhild’s measurement agrees: \displaystyle \frac{1.25}{3} = \frac{5}{12}, since 1.25 \times 12 = 15 = 3 \times 5.
(a) For the ridge, call the height h:
The ridge is 3 m above the eave.
(b) For the vent, call the distance in r:
The vent goes 4.8 m in from the eave. The pitch has no unit: it is a ratio, so it holds in meters or in feet alike.
Three points that look lined up
Keoni plots three points on a grid, as in Figure 4. From P he goes 5 right and 3 up to reach Q, and from Q he goes 8 right and 5 up to reach R. On his drawing the three points look lined up. Are they on one line?
By the test for a bend, compare the ratios of the two slope triangles:
The ratios differ, so P, Q and R are not on one line. The path bends at Q, very slightly: carrying on from Q at the steepness of PQ, 8 across would mean \displaystyle 8 \times \frac{3}{5} = 4.8 up, and R is 5 up, just 0.2 of a square higher. The eye cannot judge that; the ratios can. The same test with coordinates is the three-point test.
Boundaries
The line must not be vertical. A vertical line has no slope triangle: two of its points are straight above each other, the run is 0, the triangle flattens into a segment, and rise over run would divide by zero. A vertical line has no slope.
A horizontal line has no slope triangle either. Its rise is 0 between every pair of points, so its slope is 0 everywhere, but there is no triangle to draw.
The legs must be horizontal and vertical. A right triangle with its hypotenuse on the line but its legs tilted is not a slope triangle, and its legs can be in a quite different ratio. The proof used the horizontal legs to get parallel lines and so equal angles; tilted legs give no such angles.
The line must be straight. On a curve, triangles drawn under different pieces need not be similar, and their ratios change from place to place. That is a rate of change that is not constant.
Equal ratios need a shared point. Equal ratios from P to Q and from Q to R, going up both times or down both times, put all three points on one line, because the second triangle starts at Q, which is already on the line. Equal ratios from P to Q and from R to S, with no point shared, can belong to two different parallel lines.
Where this goes next
Because every slope triangle of a line gives the same ratio, the triangle from a fixed point of the line to any other point (x, y) on it gives that ratio too. Writing that down gives the equation of the line: slope-intercept form.
The same argument with two different lines shows that lines with equal slopes are parallel, and turning a slope triangle through a right angle shows what happens to the slope of a perpendicular line: parallel lines and perpendicular slopes.
The fact underneath this lesson, that the ratio of two sides of a right triangle depends only on its angles, is where trigonometry starts, beginning with the sine ratio.
Exercises
Slope triangles on a grid
- In Figure 1, draw the slope triangle on AC, below the line. Give its run and rise, the scale factor that turns the small triangle into it, and the slope it gives.
- In Figure 3, a slope triangle is drawn on PQ above the line instead of below it. What are its horizontal and vertical legs, and what shape do the two triangles on PQ make together?
- In Figure 3, draw the slope triangle on PS, below the line. Give its legs, the scale factor from the small triangle on PQ, and the slope it gives.
Finding a missing leg
- A slope triangle of a line has run 4 and rise 7. Another slope triangle of the same line has run 10. Find its rise.
- A slope triangle of a line has run 9 and rise 4. Another has rise 10. Find its run.
- A line falls to the right. One of its slope triangles has a horizontal leg of 8 and a vertical leg of 2. Another has a vertical leg of 5. Find its horizontal leg, and the slope of the line.
- Two slope triangles of one line have run 4 and rise x, and run 10 and rise 9. Find x.
On one line or not
- From P go 2 right and 7 up to reach Q, then from Q go 6 right and 21 up to reach R. Are P, Q and R on one line?
- From P go 7 right and 2 up to reach Q, then from Q go 10 right and 3 up to reach R. Are P, Q and R on one line?
The roof
- A straight roof edge has a 4 in 12 pitch. (a) How far above the eave is it at 9 m in from the eave, measured horizontally? (b) How far in from the eave is the edge 2.5 m above the eave?
- Radek measures a roof edge from the eave. At 2 m in, the edge is 0.75 m above the eave; at 6.4 m in, it is 2.4 m above. Are the eave and the two measured points on one straight line? If so, write the pitch as a number “in 12”.
More points on a line
- A line passes through A(1, -2) and B(4, 5). (a) Point C on the line has x-coordinate 10. Find its y-coordinate. (b) Point D on the line has y-coordinate -16. Find its x-coordinate.
Reasoning
- The proof showed two pairs of equal angles in two slope triangles. Explain why the third angles, where the line meets the vertical legs, are equal as well, and name the parallel lines that give a second way to see it.
- The points P(0, 0), Q(2, 1), R(0, 3) and S(2, 4) give a slope triangle on PQ and one on RS, both with run 2 and rise 1. Are all four points on one line? Explain why equal ratios settled the question in Exercise 8 but do not settle it here.
- Ingeborg says that two slope triangles of the same line must be congruent. Give an example that shows she is wrong, and say what is true instead.
- Explain why no slope triangle can be drawn for a vertical line, and why a horizontal line has the same slope everywhere even though it has no slope triangle.
Answers
- Run 12, rise 8, scale factor 4, slope \displaystyle \frac{2}{3}. From A to C the run is 3 + 9 = 12 and the rise is 2 + 6 = 8; \displaystyle \frac{12}{3} = \frac{8}{2} = 4 and \displaystyle \frac{8}{12} = \frac{2}{3}.
- Horizontal leg 3 and vertical leg 1, the same as the triangle below. Together the two triangles make a 3 by 1 rectangle with PQ as its diagonal.
- Horizontal leg 15 and vertical leg 5, scale factor 5, slope \displaystyle -\frac{1}{3}. From P to S the line goes 15 across and 5 down; \displaystyle \frac{15}{3} = \frac{5}{1} = 5, and \displaystyle \frac{5}{15} = \frac{1}{3}, with a minus sign because the line falls.
- 17.5. \displaystyle \frac{h}{10} = \frac{7}{4}, so \displaystyle h = \frac{70}{4} = 17.5.
- 22.5. \displaystyle \frac{10}{r} = \frac{4}{9}, so \displaystyle r = \frac{90}{4} = 22.5. The rise was scaled by 2.5, and 9 \times 2.5 = 22.5.
- Horizontal leg 20, slope \displaystyle -\frac{1}{4}. The vertical leg was scaled by \displaystyle \frac{5}{2} = 2.5, so the horizontal leg is 8 \times 2.5 = 20. The ratio is \displaystyle \frac{2}{8} = \frac{1}{4}, negative because the line falls.
- x = 3.6. \displaystyle \frac{x}{4} = \frac{9}{10}, so \displaystyle x = \frac{36}{10} = 3.6.
- Yes. \displaystyle \frac{7}{2} = 3.5 and \displaystyle \frac{21}{6} = 3.5. The second triangle is the first scaled by 3, so the point 6 right of Q on line PQ is 21 up from Q, which is R.
- No. \displaystyle \frac{2}{7} \approx 0.286 and \displaystyle \frac{3}{10} = 0.3. The ratios differ, so the slope triangles are not similar.
- (a) 3 m. \displaystyle \frac{h}{9} = \frac{4}{12}, so \displaystyle h = \frac{36}{12} = 3. (b) 7.5 m. \displaystyle \frac{2.5}{r} = \frac{4}{12}, so \displaystyle r = \frac{2.5 \times 12}{4} = 7.5.
- Yes, and the pitch is 4.5 in 12. \displaystyle \frac{0.75}{2} = 0.375 and \displaystyle \frac{2.4}{6.4} = 0.375. Both triangles start at the eave, so equal ratios put all three points on one line. A run of 12 gives a rise of 12 \times 0.375 = 4.5.
- (a) y = 19. The slope triangle on AB has run 3 and rise 7. From B to C the run is 10 - 4 = 6, twice 3, so the rise is 2 \times 7 = 14 and y = 5 + 14 = 19. (b) x = -5. From D up to A the rise is -2 - (-16) = 14, twice 7, so the run is 2 \times 3 = 6. The line goes up to the right, so D is 6 to the left of A: x = 1 - 6 = -5.
- In each triangle the angles add to 180^\circ (Triangle angle sum). Two angles of one triangle equal two angles of the other, so the third angles are 180^\circ minus the same total and are equal. A second way: the vertical legs lie on vertical lines, which are parallel, and the line crosses both, so the two top angles are corresponding angles on those parallel lines.
- No. R(0, 3) is 3 straight above P(0, 0), so it is not on the line through P and Q: PQ and RS lie on two parallel lines. In Exercise 8 the second triangle starts at Q, which is on line PQ, and has that line’s ratio, so it matches the slope triangle of line PQ from Q and ends on the line, at R. Here it starts at R, off the line, so equal ratios say only that the two lines are equally steep.
- In Figure 1 the slope triangles with legs 3 and 2 and with legs 9 and 6 lie on one line but are different sizes, so they are not congruent. What is true is that they are similar: the same shape, with every side scaled by one factor, here 3.
- Any two points of a vertical line are straight above each other, so the run is 0: the triangle collapses to a segment and rise over run would divide by 0. On a horizontal line every pair of points has rise 0, so the slope is \displaystyle \frac{0}{\text{run}} = 0 for every pair.