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Geometry

Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

$Math Studyset Vaia Explanations$ Math

Student Edition · 227 pages

ISBN: 9780395977279

Chapter 4: Q32. (page 139)

Draw an isosceles triangle and then join the midpoints of its sides to form another triangle. What can you deduce about this second triangle? Explain.

Short Answer

Expert verified

The second triangle is also an isosceles triangle.

Step by step solution

01

– Draw a diagram.

A figure shows $Δ A B C$ is an isosceles triangle with $\bar{A B} ≅ \bar{A C}$ and $D, E, F$ are the mid points of sides, $A B, A C$ and $B C$ respectively.

By joining the midpoints of sides of $Δ A B C$ another $Δ D E F$ is formed.

02

– Description of step.

A line segment which joins the midpoints of two sides of the triangle has a length equal to half the length of the third side.

Therefore, from the figure we can conclude that

$\begin{matrix} \bar{D E} = \frac{1}{2} \bar{B C} \\ \bar{E F} = \frac{1}{2} \bar{A B} \\ \bar{D F} = \frac{1}{2} \bar{A C} \end{matrix}$

03

– Description of step.

As $Δ A B C$ is an isosceles triangle, $\bar{A B} = \bar{A C}$ . Multiply both the sides of $\bar{A B} = \bar{A C}$ by $\frac{1}{2}$ .

$\begin{matrix} \frac{1}{2} \bar{A B} = \frac{1}{2} \bar{A C} \\ \bar{E F} = \bar{D F} \end{matrix}$

Therefore, by definition of an isosceles triangle, $Δ D E F$ is also an isosceles triangle with $\bar{E F} ≅ \bar{D F}$ .

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Most popular questions from this chapter

Suppose you know that $\bar{D B} ⊥ \bar{D C} .$ Explain how you can deduce that $\bar{D B} ⊥ \bar{B A} .$

For the following figure, does the SAS postulates justify that the two triangles are congruent.

In the following figure, the two-triangle shown are congruent. Then explain the following statement.

Deduce that $O$ is the midpoint of any segment.

Given: $Δ ABC$ in plane M; D not in plane M;

$∠ ACB ≅ ∠ ABC; ∠ DCB ≅ ∠ DBC$

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See all solutions

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