Question

part 1 of 8
write a paragraph proof using the figure shown to prove that ab + cb > ac.
to prove that ab + cb > ac, start by identifying relationships between line segments in the given figure. the segment \\( \overline{bd} \\) is
\\( \boldsymbol{\text{the centroid}} \\)
\\( \boldsymbol{\text{the orthocenter}} \\)
\\( \boldsymbol{\text{an altitude}} \\)
\\( \boldsymbol{\text{an angle bisector}} \\)
\\( \boldsymbol{\text{a triangle bisector}} \\)
\\( \boldsymbol{\text{of } \triangle abc} \\)

Explanation:

Step1: Use the property of right - angled triangles

In right - angled triangle \(ABD\), by the Pythagorean theorem, \(AB=\sqrt{AD^{2}+BD^{2}}\), so \(AB > AD\).

Step2: Use the property of right - angled triangles

In right - angled triangle \(CBD\), by the Pythagorean theorem, \(CB=\sqrt{CD^{2}+BD^{2}}\), so \(CB>CD\).

Step3: Add the two inequalities

Since \(AB > AD\) and \(CB > CD\), then \(AB + CB>AD + CD\).

Step4: Simplify the right - hand side

Because \(AD + CD=AC\), we get \(AB + CB>AC\).

Answer:

\(AB + CB>AC\)