Is the statement that a convex quadrilateral with perpendicular and equal-length diagonals… — ACT

Is the statement that a convex quadrilateral with perpendicular and equal-length diagonals must be a square true or false?

Answer: Answer

Answer

The statement is false because perpendicular and equal-length diagonals do not guarantee a quadrilateral is a square; they must also bisect each other.

The statement is false because having perpendicular and equal-length diagonals is a necessary condition for a square, but not a sufficient one. A quadrilateral must also have diagonals that bisect each other to be a square.

Step-by-Step Solution

Identify the properties of a square's diagonals.

In a square, the diagonals are perpendicular, equal in length, and bisect each other.

To evaluate the statement, we must compare the given diagonal conditions with the complete set of diagonal properties of a square.

Analyze whether perpendicularity and equality alone are sufficient to define a square.

Without the bisection property, we cannot guarantee the quadrilateral is a parallelogram, which is a prerequisite for being a square.

A square is a specific type of parallelogram, so any set of sufficient conditions must first satisfy the definition of a parallelogram.

Construct a counterexample where the diagonals are perpendicular and equal in length, but do not bisect each other.

Consider a quadrilateral with vertices

A(0, 3)

B(2, 0)

C(0, -1)

, and

D(-2, 0)

on a standard coordinate plane. The diagonal

AC

has a length of

4

along the

y

-axis, and the diagonal

BD

has a length of

4

along the

x

-axis. They intersect at the origin

(0, 0)

at a right angle.

Providing a single counterexample is sufficient to prove that the statement is false.

Verify if the constructed quadrilateral is a square.

The side lengths are

AB = \sqrt{13}

and

BC = \sqrt{5}

. Since the sides are not equal, this quadrilateral is a kite, not a square.

This confirms that a quadrilateral can have perpendicular and equal-length diagonals without being a square.

Key Concept

Diagonal properties of quadrilaterals

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