Question

and s is the same as the slope between s and t, given that triangles a and b are similar.
definition of slope
$5 - 3 = 15 - 9$
the area of triangle a is greater than the area of triangle b.
$\frac{5}{3} = $ slope

Explanation:

To determine the correct statement related to the slope and similar triangles:

Step 1: Recall the slope formula

The slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(\frac{y_2 - y_1}{x_2 - x_1}\). For similar triangles, the ratio of vertical change to horizontal change (slope) is constant.

Step 2: Analyze Triangle A

For Triangle A (with points \(R\), \(O\), and the right angle), the vertical change is \(5\) and horizontal change is \(3\). So slope \(= \frac{5}{3}\).

Step 3: Analyze Triangle B

For Triangle B (with points \(S\), \(T\), and the right angle), the vertical change is \(15\) and horizontal change is \(9\). Simplifying \(\frac{15}{9} = \frac{5}{3}\), which matches the slope of Triangle A.

Step 4: Evaluate the options

• "Definition of slope": Relevant, as slope is defined by \(\frac{\text{vertical change}}{\text{horizontal change}}\), and similar triangles preserve this ratio.
• "\(5 - 3 = 15 - 9\)": \(5 - 3 = 2\), \(15 - 9 = 6\) (not equal), so this is false.
• "The area of triangle \(A\) is greater than the area of triangle \(B\)": Area of \(A = \frac{1}{2}(3)(5) = 7.5\), Area of \(B = \frac{1}{2}(9)(15) = 67.5\) (so \(A\) is smaller), this is false.
• "\(\frac{5}{3} = \text{slope}\)": True, as shown by Triangle A’s slope (and confirmed by Triangle B’s slope).

If the task is to identify the correct statement, the valid ones are:

• "Definition of slope" (explains why slopes are equal for similar triangles).
• \(\boldsymbol{\frac{5}{3} = \text{slope}}\) (matches the slope calculation for both triangles).

For the false statements:

• \(5 - 3 = 15 - 9\) is false ( \(2

eq 6\) ).

• "The area of triangle \(A\) is greater than the area of triangle \(B\)" is false ( \(7.5 < 67.5\) ).

Answer:

To determine the correct statement related to the slope and similar triangles:

Step 1: Recall the slope formula

The slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(\frac{y_2 - y_1}{x_2 - x_1}\). For similar triangles, the ratio of vertical change to horizontal change (slope) is constant.

Step 2: Analyze Triangle A

For Triangle A (with points \(R\), \(O\), and the right angle), the vertical change is \(5\) and horizontal change is \(3\). So slope \(= \frac{5}{3}\).

Step 3: Analyze Triangle B

For Triangle B (with points \(S\), \(T\), and the right angle), the vertical change is \(15\) and horizontal change is \(9\). Simplifying \(\frac{15}{9} = \frac{5}{3}\), which matches the slope of Triangle A.

Step 4: Evaluate the options

• "Definition of slope": Relevant, as slope is defined by \(\frac{\text{vertical change}}{\text{horizontal change}}\), and similar triangles preserve this ratio.
• "\(5 - 3 = 15 - 9\)": \(5 - 3 = 2\), \(15 - 9 = 6\) (not equal), so this is false.
• "The area of triangle \(A\) is greater than the area of triangle \(B\)": Area of \(A = \frac{1}{2}(3)(5) = 7.5\), Area of \(B = \frac{1}{2}(9)(15) = 67.5\) (so \(A\) is smaller), this is false.
• "\(\frac{5}{3} = \text{slope}\)": True, as shown by Triangle A’s slope (and confirmed by Triangle B’s slope).

If the task is to identify the correct statement, the valid ones are:

• "Definition of slope" (explains why slopes are equal for similar triangles).
• \(\boldsymbol{\frac{5}{3} = \text{slope}}\) (matches the slope calculation for both triangles).

For the false statements:

• \(5 - 3 = 15 - 9\) is false ( \(2

eq 6\) ).

• "The area of triangle \(A\) is greater than the area of triangle \(B\)" is false ( \(7.5 < 67.5\) ).