Track Meet Results Board: Why Slope Stays Constant

Use similar triangles to build and explain linear equations.

Lap Challenge Points

The results board includes a lap challenge. Its line models points y earned for x completed laps. Use the two right triangles shown.

Find the slope of line OQ using triangle ORP. Write a fraction.

Slope =

Why are triangles ORP and OSQ similar?

A. They have right angles and equal hypotenuse lengths.B. They have equal perimeters and equal vertical legs.C. They have right angles and the same angle at O.D. They have equal areas and equal horizontal legs.

Which proportion compares corresponding vertical and horizontal legs of the two triangles?

A. 2/4 = 6/3B. 2/3 = 6/4C. 2/3 = 4/6D. 3/2 = 4/6

Let (x, y) be any point on the modeled line with x > 0. A right triangle from O to this point has horizontal leg x and vertical leg y. Which derivation gives the equation?

A. y/x = 3/2, so y = (3/2)x.B. y/x = 2/3, so y = (2/3)x.C. y/x = 2/3, so y = x + 2/3.D. x/y = 2/3, so y = (3/2)x.

A Shifted Results Line

For a separate meet challenge, the board converts a raw score difference x into an adjusted score difference y. Both differences are signed numbers. Use the graph.

Use triangle BHC to find the slope of line AC.

Slope =

A new right triangle is drawn between B and another point (x, y) on the line, where x > 0. Which expression is its vertical change divided by its horizontal change?

A. (y + 3)/xB. y/xC. x/(y − 3)D. (y − 3)/x

The new triangle is similar to triangle BHC, so (y − 3)/x = 1/2. Which equation follows?

A. y = 3x + 1/2B. y = (1/2)x + 3C. y = (1/2)(x + 3)D. y = (1/2)x − 3

The board uses the same line for scores outside the displayed window. What adjusted score difference does it show for a raw score difference of −10?

Adjusted score difference = points

For any nonvertical line crossing the y-axis at (0, b), a slope triangle to (x, y), with x ≠ 0, gives m = (y − b)/x. Which algebraic step produces y = mx + b?

A. Multiply by x, then add b to both sides.B. Divide by x, then add b to both sides.C. Multiply by b, then add x to both sides.D. Multiply by x, then subtract b from both sides.

Check the Board Models

Apply slope triangles and linear equations to other features of the track meet results board.

10.

A replay display models a runner moving at a constant speed. At x = 0 seconds, the runner has traveled y = 0 meters. A slope triangle has a horizontal change of 4 seconds and a vertical change of 28 meters. Which equation models the distance?

A. y = 7xB. y = 28xC. y = 4x + 28D. y = (1/7)x

11.

Another replay starts when a runner is already 15 meters from the starting line. Its straight-line model has a slope triangle with horizontal change 3 seconds and vertical change 18 meters. How far from the starting line is the runner 8 seconds after this replay begins?

Distance = meters

12.

A straight line on the board shows the time remaining before the next race. It passes through (0, 12), where x is elapsed time in minutes and y is time remaining in minutes. One slope triangle has a horizontal change of 4 and a signed vertical change of −4. A larger triangle along the same line has a horizontal change of 9. Find its signed vertical change.

Signed vertical change = minutes

13.

Nia draws two right slope triangles on the same nonvertical results-board line. Their horizontal legs are parallel to the x-axis, and their vertical legs are parallel to the y-axis. She says, “The larger triangle has a longer vertical leg, so that part of the line is steeper.” Explain why her conclusion is incorrect using angles, similarity, and leg ratios.