Question

practice problems
7 problems
1 here are triangles abc and abc.
what is the measure of angle abc?
a 20° b 40°
c 60° d 80°
2 triangle def is formed by connecting the midpoints of the sides of triangle abc.
the lengths of the sides of def are shown.
what is the length of ab?

Explanation:

Step1: Check the similarity of triangles

We can observe that \(\frac{BB'}{BC}=\frac{3}{3 + 6}=\frac{1}{3}\), \(\frac{AA'}{AC}=\frac{2}{2+4}=\frac{1}{3}\). By the Side - Angle - Side (SAS) similarity criterion (the included angle \(\angle BAC\) is common), \(\triangle ABB'\sim\triangle ABC\).

Step2: Use the property of similar triangles

For similar triangles, corresponding angles are equal. Since \(\angle ABC = 40^{\circ}\) and \(\angle A'B'C\) corresponds to \(\angle ABC\) (because of the similarity of the triangles formed by the line segments \(A'B'\) and \(AB\), \(B'C\) and \(BC\), \(A'C\) and \(AC\)), we have \(\angle A'B'C=\angle ABC\).

Step1: Use the mid - segment theorem

The mid - segment theorem states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. In \(\triangle ABC\), \(DF\) is a mid - segment.

Step2: Calculate the length of \(AB\)

Since \(DF\) is a mid - segment and \(DF = 4\), and by the mid - segment theorem \(DF=\frac{1}{2}AB\). Then \(AB = 2\times DF\).

Answer:

B. \(40^{\circ}\)

For the second problem (assuming we are still in the Geometry sub - field of Mathematics):