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Chapter 1: Problem 7

Prove that the diagonals of a parallelogram bisect each other.

Short Answer

Expert verified
Using properties of parallelograms and the Side-Angle-Side (SAS) congruence, we prove that corresponding triangles created by diagonals AC and BD are congruent. Specifically, \(\triangle ABE \cong \triangle DEC\) and \(\triangle AED \cong \triangle CEB\). It follows that the diagonals of a parallelogram bisect each other at point E.

Step by step solution

01

Label the parallelogram

Let's label the parallelogram ABCD, with A and C being opposite vertices and B and D being the other opposite vertices. Let the diagonals AC and BD intersect at point E.
02

Determine the properties we know about parallelograms

In a parallelogram, opposite sides are parallel and congruent. So, we have: 1. \(AB \parallel CD\) 2. \(BC \parallel AD\) 3. \(AB = CD\) 4. \(BC = AD\)
03

Prove that two sets of corresponding triangles are congruent

We are going to prove that two sets of corresponding triangles are congruent by Side-Angle-Side (SAS) congruence. Consider triangles ABE and CDE. We have: 1. \(AB = CD\) (from the properties of parallelograms) 2. \(AE = CE\) (E is the midpoint of diagonal AC) 3. \(\angle AEB \cong \angle DEC\) (alternate interior angles are congruent since \(AB \parallel CD\)) So, by SAS congruence, we have: \(\triangle ABE \cong \triangle DEC\) Similarly, consider triangles AED and CEB. We have: 1. \(BC = AD\) (from the properties of parallelograms) 2. \(DE = BE\) (E is the midpoint of diagonal BD) 3. \(\angle EAD \cong \angle ECB\) (alternate interior angles are congruent since \(BC \parallel AD\)) So, by SAS congruence, we have: \(\triangle AED \cong \triangle CEB\)
04

Show that the diagonals bisect each other

Since \(\triangle ABE \cong \triangle DEC\), we have \(EB = EC\). This means that diagonal BD is bisected at point E. Similarly, since \(\triangle AED \cong \triangle CEB\), we have \(AE = CE\). This means that diagonal AC is bisected at point E. Thus, we have proved that the diagonals of a parallelogram bisect each other.

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Most popular questions from this chapter

Show that the matrices $$ \left(\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right), \quad\left(\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right), \quad\left(\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right), \quad \text { and } \quad\left(\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right) $$ generate \(\mathrm{M}_{2 \times 2}(F)\).

Let \(S=\\{(1,1,0),(1,0,1),(0,1,1)\\}\) be a subset of the vector space \(\mathrm{F}^{3}\). (a) Prove that if \(F=R\), then \(S\) is linearly independent. (b) Prove that if \(F\) has characteristic two, then \(S\) is linearly dependent.

Let \(f, g, \in \mathcal{F}(R, R)\) be the functions defined by \(f(t)=e^{r t}\) and \(g(t)=e^{s t}\), where \(r \neq s\). Prove that \(f\) and \(g\) are linearly independent in \(\mathcal{F}(R, R)\).

Recall from Example 3 in Section \(1.3\) that the set of diagonal matrices in \(\mathrm{M}_{2 \times 2}(F)\) is a subspace. Find a linearly independent set that generates this subspace.

Find \(k\) so that \(u\) and \(v\) are orthogonal, where: (a) \(u=(3, k,-2), v=(6,-4,-3)\) (b) \(u=(5, k,-4,2), v=(1,-3,2,2 k)\) (c) \(u=(1,7, k+2,-2), v=(3, k,-3, k)\)
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