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Chapter 5: Q34 (page 194)

Prove that if the diagonals of parallelograms are perpendicular, then the parallelogram is a rhombus.

Short Answer

Expert verified

If the diagonals of parallelograms are perpendicular, then the parallelogram is a rhombus.

Step by step solution

01

Step 1. Check the figure.

Consider the figure.

02

Step 2. Step description.

The CPCTC theorem states that when two triangles are congruent, then every corresponding part of one triangle is congruent to the other.

03

Step 3. Step description.

From the figure, 螖AODand 螖AOB.

OA=OA (Sides are common)

As the diagonals are perpendicular such that 鈭�AOB鈮�鈭�AOD=90鈭�and the diagonals of a parallelogram bisect such that OB=OD.

Therefore, by the side-angle-side postulate 螖BOA鈮�螖BOA.

Now, by CPCT theorem AD鈫�鈮�AB鈫�.

As the opposite sides are congruent in parallelogram thus

AB鈫�鈮�DC鈫�&AD鈫�鈮�BC鈫�鈥�鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌�鈭�AB鈫�鈮�AD鈫�鈬�AD鈫�鈮�DC鈫�鈮�AB鈫�鈮�BC鈫�鈥�

Therefore, the parallelogram is a rhombus.

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Most popular questions from this chapter

Study the markings on each figure and decide whether ABCD must be a parallelogram. If the answer is yes, state the definition or theorem that applies.

Each figure in Exercises 19-24 is a parallelogram with its diagonals drawn. Find the values of x and y.

Study the markings on each figure and decide whether ABCD must be a parallelogram. If the answer is yes, state the definition or theorem that applies.

In Exercise 4 quad. ABCD is a parallelogram. Find the values of x, y, and z.

The quadrilaterals numbered 1, 2, 3, 4, and 5 are parallelograms. If you wanted to show that quadrilateral 6 is also a parallelogram, which of the five methods listed on page 172 would be easiest to use?
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