Question

part a
explain how you know that the triangles are similar.
all right triangles are similar.
\\( \overline { a d } \\) and \\( \overline { a b } \\) lie on the same line, as do \\( \overline { a e } \\) and \\( \overline { a c } \\), so they are similar by sss.
\\( \triangle a b c \\) is transformed into \\( \triangle a c d \\) using a dilation of \\( \frac { 1 } { 2 } \\), so they are similar.
angle \\( a \\) is the same for both triangles and \\( m ? c \\) is equal to \\( m ? e \\), so the triangles are similar by \\( a a \\) similarity.
part b
how does knowing the triangles are similar help you show that the slope of the line through the hypotenuses is constant?
the lengths of the corresponding sides add to a constant value, so the length of the hypotenuse is also constant.
all similar triangles are oriented in the same way, so the slope of the hypotenuse must always be the same.
\\( \angle a \\) is shared by both triangles, so that means \\( \overline { a b } = \overline { a d } \\).
the lengths of the corresponding sides are proportional, so \\( \frac { b c } { a c } = \frac { d e } { a b } \\). each of these ratios is also equal to the slope of the line, so the slope is constant.

Explanation:

Part A

Step1: Analyze similarity criteria

• For two triangles to be similar, we can use AA (Angle - Angle) similarity. In \(\triangle ABC\) and \(\triangle ADE\), \(\angle A\) is common.
• Since \(BC\parallel DE\) (because they are vertical and horizontal segments in a coordinate - plane context, and the lines are parallel), \(\angle ACB=\angle AED\) (corresponding angles).
• AA similarity states that if two angles of one triangle are equal to two angles of another triangle, the triangles are similar.

Part B

Step1: Recall slope formula and similarity properties

• The slope formula is \(m = \frac{\text{rise}}{\text{run}}\). For \(\triangle ABC\), the slope of \(AB\) (where \(AB\) is the hypotenuse) is \(m_1=\frac{BC}{AC}\). For \(\triangle ADE\), the slope of \(AD\) (where \(AD\) is the hypotenuse) is \(m_2=\frac{DE}{AE}\).
• Because \(\triangle ABC\sim\triangle ADE\) (by AA similarity), the ratios of corresponding sides are equal. That is, \(\frac{BC}{AC}=\frac{DE}{AE}\) (by the property of similar triangles: \(\frac{\text{side}_1}{\text{side}_2}=\frac{\text{corresponding side}_1}{\text{corresponding side}_2}\))

Answer:

• Part A: Angle \(A\) is the same for both triangles and \(m\angle C\) is equal to \(m\angle E\), so the triangles are similar by \(AA\) similarity.
• Part B: The lengths of the corresponding sides are proportional, so \(\frac{BC}{AC}=\frac{DE}{AE}\). Each of these ratios is also equal to the slope of the line, so the slope is constant.