Question

given a line segment with length a, what special parallelograms meet the following condition? show each sketch. both diagonals have length a.
□ a. rhombus
□ b. rectangle
□ c. all parallelograms
□ d. square

Explanation:

Step1: Recall properties of parallelograms

- **Rhombus**: Diagonals are not equal in length (they bisect each other at right angles, but lengths differ unless it's a square).
- **Rectangle**: Diagonals are equal in length (property of rectangles: diagonals of a rectangle are congruent).
- **All parallelograms**: In general parallelograms, diagonals bisect each other but are not necessarily equal. Only rectangles (and squares, which are special rectangles) have equal diagonals among parallelograms.
- **Square**: A square is a special rectangle (and rhombus), so its diagonals are equal. But we need to check which parallelograms have diagonals of length \(a\) (equal length).

Step2: Analyze each option

- **Option A (Rhombus)**: Diagonals of a rhombus are not equal unless it's a square. So a general rhombus does not have equal diagonals. Eliminate A.
- **Option B (Rectangle)**: By property of rectangles, diagonals are equal. So if a rectangle is a parallelogram with diagonals of length \(a\), it satisfies the condition.
- **Option C (All parallelograms)**: As mentioned, most parallelograms (like rhombus, non - rectangular parallelograms) do not have equal diagonals. So this is incorrect.
- **Option D (Square)**: A square is a special case of a rectangle (and rhombus). While a square has equal diagonals, the set of rectangles (including squares) is a more general set that satisfies the "diagonals equal" property among parallelograms. But we need to see which option is correct. Wait, actually, the question is about which special parallelograms have both diagonals of length \(a\) (i.e., equal diagonals). The property of equal diagonals in parallelograms is a defining property of rectangles (and squares, which are rectangles). So rectangles (including squares) have equal diagonals. But let's re - check:

Wait, the key is: In a parallelogram, if diagonals are equal, then it is a rectangle (theorem). So the parallelograms with equal diagonals are rectangles (and squares, which are special rectangles). So among the options, rectangle (B) and square (D) have equal diagonals. But let's check the options again. Wait, the problem says "what special parallelograms meet the following condition? Both diagonals have length \(a\)".

Wait, maybe I misread. Let's re - express:

- Rhombus: diagonals are perpendicular bisectors, not equal (unless square).
- Rectangle: diagonals are equal (by definition of rectangle: a parallelogram with right angles, and diagonals are equal).
- All parallelograms: No, only rectangles (and squares) in parallelograms have equal diagonals.
- Square: A square is a rectangle with all sides equal. So square is a special rectangle.

So the parallelograms with equal diagonals are rectangles (and squares). So among the options, B (Rectangle) and D (Square) have equal diagonals. But let's check the original problem's options. Wait, maybe the question is which of the given options (A - D) are parallelograms with equal diagonals.

Wait, the correct answer: The property of a parallelogram having equal diagonals is equivalent to it being a rectangle (including square). So:

- Rectangle (B): Yes, diagonals are equal.
- Square (D): Yes, diagonals are equal (since square is a rectangle).
- Rhombus (A): No.
- All parallelograms (C): No.

But maybe the question is a multiple - choice where we have to select the correct ones. Wait, the user's problem has checkboxes for A, B, C, D. Let's re - evaluate:

The theorem: In a parallelogram, if the diagonals are equal, then the parallelogram is a rectangle. So the set of parallelograms with equal diagonals is the set of rectangles (and squares, which are rectangles). So rectangles (B) and squares (D) are parallelograms with equal diagonals. But let's check the options again.

Wait, maybe the question is asking which of the given figures (A: Rhombus, B: Rectangle, C: All parallelograms, D: Square) have both diagonals of length \(a\) (i.e., equal diagonals). So:

- A: Rhombus: diagonals not equal (unless square) → no.
- B: Rectangle: diagonals equal → yes.
- C: All parallelograms: no, only rectangles (and squares) → no.
- D: Square: diagonals equal (since square is a rectangle) → yes.

But maybe the intended answer is B (Rectangle) and D (Square). But let's see the options. Wait, maybe the problem is from a source where rectangle is the answer (since square is a special rectangle). Alternatively, maybe I made a mistake. Let's recall:

In a parallelogram, diagonals are equal if and only if it is a rectangle. So the special parallelograms with equal diagonals are rectangles (and squares, which are rectangles). So among the options, B (Rectangle) and D (Square) are correct. But let's check the original problem's options again. If we have to choose from A, B, C, D, the ones with equal diagonals are B (Rectangle) and D (Square). But maybe the question is designed to have B and D as correct, but let's see the initial analysis.

Wait, maybe the user's problem is a multiple - choice where we need to select the correct option(s). Let's assume that the options are checkboxes, and we need to find which ones are correct.

So:

- A (Rhombus): No.
- B (Rectangle): Yes.
- C (All parallelograms): No.
- D (Square): Yes.

But maybe the question is considering that a square is a special case of a rectangle, and the main category is rectangle. But according to the theorem, in a parallelogram, equal diagonals imply it is a rectangle. So the correct options are B (Rectangle) and D (Square).

Answer:

B. Rectangle, D. Square