Question

triangle tuv is similar to triangle xyz. which of the statements is true?
o a. the slope of uv is equal to the slope of xy
o b. the slope of tu is equal to the slope of yz
o c. the slope of vt is equal to the slope of zx
o d. the slope of uv is equal to the slope of zx

Explanation:

Step1: Recall the slope formula

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For similar triangles, the ratios of the corresponding vertical and horizontal sides are equal. The slope of a line segment in a coordinate - plane is determined by the change in \(y\) over the change in \(x\).

Step2: Analyze the line segments

• For \(\overline{VT}\) and \(\overline{ZX}\):
• Let's assume coordinates. For example, if we consider the construction of the slope using the right - triangle sides. The slope of \(\overline{VT}\) is calculated as \(\frac{\text{vertical change (in triangle TUV)}}{\text{horizontal change (in triangle TUV)}}\), and the slope of \(\overline{ZX}\) is calculated as \(\frac{\text{vertical change (in triangle XYZ)}}{\text{horizontal change (in triangle XYZ)}}\).
• Since \(\triangle TUV\sim\triangle XYZ\), the ratio \(\frac{\text{vertical change}}{\text{horizontal change}}\) is the same for \(\overline{VT}\) (using the sides of \(\triangle TUV\)) and \(\overline{ZX}\) (using the sides of \(\triangle XYZ\)).

Step3: Check other options

• Option A: \(\overline{UV}\) is a horizontal - like side in \(\triangle TUV\) (assuming coordinate - based side - orientation from the graph), and \(\overline{XY}\) is a vertical - like side in \(\triangle XYZ\). Their slopes are not equal. The slope of a horizontal - like segment (if \(y_2=y_1\)) is \(m = 0\), and the slope of a vertical - like segment (if \(x_2 = x_1\)) is undefined.
• Option B: \(\overline{TU}\) is a vertical - like side in \(\triangle TUV\) and \(\overline{YZ}\) is a horizontal - like side in \(\triangle XYZ\). Their slopes are not equal. The slope of a vertical - like segment (if \(x_2 = x_1\)) is undefined, and the slope of a horizontal - like segment (if \(y_2=y_1\)) is \(m = 0\).
• Option D: \(\overline{UV}\) is a horizontal - like side in \(\triangle TUV\) (slope \(m = 0\)) and \(\overline{ZX}\) is a non - horizontal, non - vertical side (slope \(m

eq0\)).

Answer:

C. The slope of \(\overline{VT}\) is equal to the slope of \(\overline{ZX}\)