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PSAT Math : Rhombuses

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Example Question #1 : Rhombuses

If the area of a rhombus is 24 and one diagonal length is 6, find the perimeter of the rhombus.

Possible Answers:

Correct answer:

Explanation:

The area of a rhombus is found by

A = 1/2(d₁)(d₂)

where d₁ and d₂ are the lengths of the diagonals. Substituting for the given values yields

24 = 1/2(d₁)(6)

24 = 3(d₁)

8 = d₁

Now, use the facts that diagonals are perpendicular in a rhombus, diagonals bisect each other in a rhombus, and the Pythagorean Theorem to determine that the two diagonals form 4 right triangles with leg lengths of 3 and 4. Since 3² + 4² = 5², each side length is 5, so the perimeter is 5(4) = 20.

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Example Question #1 : Rhombuses

Rhombus

Note: Figure NOT drawn to scale.

Calculate the perimeter of Quadrilateral ABCD in the above diagram if:

AC = 48

BD = 20

AB=BC=CD=DA

Possible Answers:

120

136

104

Insufficient information is given to answer the question.

Correct answer:

104

Explanation:

AB=BC=CD=DA , so Quadrilateral ABCD is a rhombus. Its diagonals are therefore perpendicular to each other, and the four triangles they form are right triangles. Therefore, the Pythagorean theorem can be used to determine the common sidelength of Quadrilateral ABCD .

We focus on $\Delta AXB$ . The diagonals are also each other's bisector, so

$AX = \frac{1}{2} (AC) = \frac{1}{2} \cdot 48= 24$

$XB = \frac{1}{2} (BD) = \frac{1}{2} \cdot 20=10$

By the Pythagorean Theorem,

$AB = \sqrt{(AX)^{2}+ (XB)^{2}} = \sqrt{24^{2}+10^{2}} = \sqrt{576+100} = \sqrt{676} = 26$

26 is the common length of the four sides of Quadrilateral ABCD , so its perimeter is $4 \times 26 = 104$ .

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Example Question #1 : How To Find The Area Of A Rhombus

A rhombus has a side length of 5. Which of the following is NOT a possible value for its area?

Possible Answers:

Correct answer:

Explanation:

The area of a rhombus will vary as the angles made by its sides change. The "flatter" the rhombus is (with two very small angles and two very large angles, say 2, 178, 2, and 178 degrees), the smaller the area is. There is, of course, a lower bound of zero for the area, but the area can get arbitrarily small. This implies that the correct answer would be the largest choice. In fact, the largest area of a rhombus occurs when all four angles are equal, i.e. when the rhombus is a square. The area of a square of side length 5 is 25, so any value bigger than 25 is impossible to acheive.

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Example Question #1 : Quadrilaterals

In Rhombus RHOM , $m \angle R = 60^{\circ }$ . If $\overline{HM}$ is constructed, which of the following is true about $\Delta RHM$ ?

Possible Answers:

$\Delta RHM$ obtuse and scalene

$\Delta RHM$ is acute and isosceles, but not equilateral

$\Delta RHM$ is acute and scalene

$\Delta RHM$ is acute and equilateral

$\Delta RHM$ is obtuse and isosceles, but not equilateral

Correct answer:

$\Delta RHM$ is acute and equilateral

Explanation:

The figure referenced is below.

Rhombus

Consecutive angles of a rhombus are supplementary - as they are with all parallelograms - so

$m \angle RHO= 180^{\circ} - m \angle R = 180^{\circ} - 60^{\circ} = 120^{\circ}$

A diagonal of a rhombus bisects its angles, so

$m \angle RHM = \frac{1}{2} m \angle RHO= \frac{1}{2} \cdot 120^{\circ} = 60^{\circ}$

A similar argument proves that $m \angle RMH = 60^{\circ}$ .

Since all three angles of $\Delta RHM$ measure $60^{\circ}$ , the triangle is acute. It is also equiangular, and, subsequently, equilateral.

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Example Question #2 : Quadrilaterals

In Rhombus RHOM , $m \angle H = 45 ^{\circ }$ . If $\overline{RO }$ is constructed, which of the following is true about $\Delta RHO$ ?

Possible Answers:

$\Delta RHO$ is right and scalene

$\Delta RHO$ is acute and scalene

$\Delta RHO$ is acute and equilateral

$\Delta RHO$ is right and isosceles, but not equilateral

$\Delta RHO$ is acute and isosceles, but not equilateral

Correct answer:

$\Delta RHO$ is acute and isosceles, but not equilateral

Explanation:

The figure referenced is below.

Rhombus

The sides of a rhombus are congruent by definition, so $\overline{RH} \cong \overline{HO}$ , making $\Delta RHO$ isosceles. It is not equilateral, since $m \angle H = 45 ^{\circ }$ , and an equilateral triangle must have three $60 ^{\circ}$ angles.

Also, consecutive angles of a rhombus are supplementary - as they are with all parallelograms - so

$m \angle HRM = 180^{\circ} - m \angle H = 180^{\circ} - 45 ^{\circ} = 135^{\circ}$

A diagonal of a rhombus bisects its angles, so

$m \angle HRO = \frac{1}{2} m \angle HRM = \frac{1}{2} \cdot 135 ^{\circ } = 67\frac{1}{2}^{\circ }$

Similarly, $m \angle HOR = 67\frac{1}{2}^{\circ }$

This makes $\Delta RHO$ acute.

The correct response is that $\Delta RHO$ is acute and isosceles, but not equilateral.

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PSAT Math : Rhombuses

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #1 : Rhombuses

Example Question #1 : Rhombuses

Example Question #1 : How To Find The Area Of A Rhombus

Example Question #1 : Quadrilaterals

Example Question #2 : Quadrilaterals

All PSAT Math Resources

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