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Chapter 4: Problem 17

What can you deduce? Name the theorem that supports your answer. Given: \(\overline{A M}\) is a median of \(\triangle A B C\) \(A B>A C\)

Short Answer

Expert verified
Based on the Triangle Inequality Theorem, it can be deduced that the area of \( \triangle A B M \) is greater than the area of \( \triangle A C M \)

Step by step solution

01

Identifying related properties

Firstly, recall that a median of a triangle is a line from a vertex to the midpoint of the opposite side. The median divides the triangle into two smaller triangles. Here, \( \overline{A M} \) is a median. So it divides \( \triangle A B C \) into two smaller triangles: \( \triangle A B M \) and \( \triangle A C M \). Considering the statement \( AB > AC \), it is clear that side \( AB \) of \( \triangle A B C \) is longer than side \( AC \).
02

Applying the appropriate theorem

According to the median properties, the two triangles divided by the median have equal areas if the two base sides are equal. But in this case, the bases are not equal as \( AB > AC \). So, according to the Triangle Inequality Theorem, if one side of a triangle is longer than another, then the triangle containing the longer side also has greater area.
03

Making the final deduction

Based on the given conditions and the applied theorem, we can deduce that \( \triangle A B M \)'s area is greater than \( \triangle A C M \)'s area.

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Most popular questions from this chapter

Given: \(\overline{A B} \cong \overline{A C} ; \overline{O B} \cong \overline{O C}\) a. Is it possible to prove that \(\angle A O B \cong \angle A O C ?\) b. Is it possible to prove that \(\angle A O B\) and \(\angle A O C\) are right angles?

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Draw and label a diagram. List, in terms of the diagram, what is given and what is to be proved. Then write a two-column proof. If pentagon \(A B C D E\) is equilateral and has right angles at \(B\) and \(E,\) then diagonals \(\overline{A C}\) and \(\overline{A D}\) form congruent triangles.

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