Free math worksheets

- Read the coordinates of the two marked points from the graph. - The horizontal side of the right triangle represents the run, and the vertical side represents the rise. - Compare the vertical change with the horizontal change in the same direction from \(A\) to \(B\).

From \(A = (0, 1)\) to \(B = (4, 3)\), the horizontal change is \(4 - 0 = 4\) and the vertical change is \(3 - 1 = 2\). Therefore the slope is \(m = \frac{\text{rise}}{\text{run}} = \frac{2}{4} = \frac{1}{2}\).

Run: \(4\) Rise: \(2\) Slope: \(\frac{1}{2}\)

- Recall the angle-angle criterion for triangle similarity. - Identify one angle that every right slope triangle has and another angle determined by the same line.

1. Each slope triangle has a right angle. 2. Because both triangles use the same line and horizontal segments, they also have a matching acute angle. 3. Two equal angles establish similarity by angle-angle, so the correct choice is B.

B) Each has a right angle, and each has the same acute angle made by the line with a horizontal segment.

- For each triangle, read the horizontal and vertical changes between its two marked endpoints. - Keep the direction consistent when you compare rise with run. - After simplifying both ratios, compare them rather than comparing the side lengths themselves.

a) From \(A = (0, 1)\) to \(B = (2, 2)\), the rise is \(1\) and the run is \(2\), so \(\frac{\text{rise}}{\text{run}} = \frac{1}{2}\). From \(B = (2, 2)\) to \(C = (6, 4)\), the rise is \(2\) and the run is \(4\), so \(\frac{\text{rise}}{\text{run}} = \frac{2}{4} = \frac{1}{2}\). b) The two right triangles have the same rise-to-run ratio even though they have different sizes. That common ratio is the slope of line \(l\), so \(m = \frac{1}{2}\).

a) \(\frac{1}{2}\) for triangle \(AB\) and \(\frac{1}{2}\) for triangle \(BC\). b) Both triangles give the same slope, \(m = \frac{1}{2}\).

- Compare the two runs to determine the scale factor. - Similar triangles scale corresponding vertical and horizontal legs by the same factor. - Compare rise divided by run for the two triangles.

1. The run increases from \(3\) to \(9\), a scale factor of \(3\). 2. Similar triangles scale corresponding legs by the same factor, so the rise is \(2 \cdot 3 = 6\). 3. The small slope is \(\frac{2}{3}\), and the large slope is \(\frac{6}{9} = \frac{2}{3}\).

The larger rise is \(6\) units. Both triangles have slope \(\frac{2}{3}\).

- Read the coordinates of \(A\) and \(B\) from the grid. - Keep the subtraction order consistent between the vertical and horizontal changes. - Compare what happens to both signs when you reverse the point order.

1. From \(A(2, 3)\) to \(B(6, 1)\), the vertical change is \(-2\) and the horizontal change is \(4\), so the slope is \(\frac{-2}{4} = -\frac{1}{2}\). 2. From \(B\) to \(A\), the vertical change is \(2\) and the horizontal change is \(-4\), so the slope is \(\frac{2}{-4} = -\frac{1}{2}\). 3. Reversing the point order changes the signs of both corresponding legs, so their ratio stays the same.

a) \(-\frac{1}{2}\) b) \(-\frac{1}{2}\) c) Reversing the points changes both differences' signs, leaving their ratio unchanged.

- On a horizontal segment, only the x-coordinate changes; on a vertical segment, only the y-coordinate changes. - Compare the angles in two right triangles drawn between the same line and horizontal segments. - Once the triangles are known to be similar, focus on the ratio of the vertical leg to the horizontal leg.

a) The horizontal change from \(P\) to \(Q\) is \(x_2 - x_1\), so the run is \(x_2 - x_1\). The vertical change is \(y_2 - y_1\), so the rise is \(y_2 - y_1\). b) Each slope triangle is a right triangle, and each has the same acute angle formed by line \(l\) and a horizontal segment. Therefore the triangles are similar by angle-angle similarity. c) Similar triangles have equal ratios of corresponding legs, so the rise-to-run ratio is constant along the line. Therefore \(m = \frac{\text{rise}}{\text{run}} = \frac{y_2-y_1}{x_2-x_1}\).

a) Run: \(x_2-x_1\); rise: \(y_2-y_1\). b) The right slope triangles are similar by angle-angle similarity. c) \(m = \frac{y_2-y_1}{x_2-x_1}\).

- Slope uses directed change, not just the positive lengths of the triangle's sides. - Track what happens to the y-coordinate as you move from the left point to the right point. - Keep the direction of the vertical change consistent with the direction used for the horizontal change.

a) The student used the positive length of the vertical side instead of the signed vertical change. From \(A=(-2,3)\) to \(B=(2,1)\), the y-coordinate decreases, so the rise is \(1-3=-2\), not \(2\). b) The run is \(2-(-2)=4\). Therefore \(m=\frac{-2}{4}=-\frac{1}{2}\).

a) The vertical change from \(A\) to \(B\) is negative because the line goes downward as x increases. b) \(-\frac{1}{2}\)

- A slope triangle must represent horizontal and vertical changes between points on the line. - Check which labeled points actually lie on the graphed line. - Compare the rise-to-run ratios from \(A\) to the two valid points.

1. Point \(D(3, 4)\) is not on the line, so a triangle whose diagonal runs from \(A\) to \(D\) does not measure the line's change. 2. From \(A(0, 0)\) to \(B(3, 2)\), the slope is \(\frac{2}{3}\). 3. From \(A\) to \(C(6, 4)\), the slope is \(\frac{4}{6} = \frac{2}{3}\). 4. The two valid right slope triangles are similar, so their corresponding rise-to-run ratios agree.

a) \(D\) is not on the line, so \(A\) to \(D\) cannot measure the line's slope. b) \(\frac{2}{3} = \frac{4}{6}\), so both valid slope triangles give slope \(\frac{2}{3}\).

- Start with the rise-to-run ratio of the smaller triangle. - Set the larger triangle's rise-to-run ratio equal to the smaller one's. - Connect each triangle's rise and run to coordinate differences on the line.

1. The first triangle gives slope \(\frac{6}{9} = \frac{2}{3}\). 2. Similar slope triangles have equal rise-to-run ratios, so \(\frac{r}{15} = \frac{2}{3}\). 3. Solving gives \(3r = 30\), so \(r = 10\). 4. The vertical and horizontal changes may scale, but their ratio stays constant. Therefore, the coordinate differences \(y_2-y_1\) and \(x_2-x_1\) have a constant ratio on the line.

\(r = 10\). Similar slope triangles preserve the rise-to-run ratio, which is why \(\frac{y_2-y_1}{x_2-x_1}\) gives the same slope for any two points on the line.

- Imagine a horizontal and a vertical leg joining each pair of points to form right triangles. - Compare the acute angle each triangle makes with the same line and a horizontal segment. - For similar triangles, think about what happens to the ratio of corresponding vertical and horizontal legs.

Draw a right slope triangle for each pair of points: \(A\) to \(B\), \(B\) to \(C\), and \(A\) to \(C\). Each triangle has a right angle, and each has the same acute angle made by the given line and a horizontal segment. Therefore the three triangles are similar by angle-angle similarity. In each triangle, the vertical leg is the change in y and the horizontal leg is the change in x. Similarity makes the ratio of corresponding vertical and horizontal legs the same for all three triangles. Thus \(\frac{y_2-y_1}{x_2-x_1}=\frac{y_3-y_2}{x_3-x_2}=\frac{y_3-y_1}{x_3-x_1}\). So any two of the three points give the same slope for the line.

The three right slope triangles are similar by angle-angle similarity, so their corresponding rise/run ratios are equal. Therefore any pair of the points gives the same slope.