ACT Math - Other Quadrilaterals

The sides of rectangle A measure to , , , and . Rectangle B is similar to Rectangle A. The shorter sides of rectangle B measure each. How long are the longer sides of Rectangle B?

Explanation

In similar rectangles, the ratio of the sides must be equal.

To solve this question, the following equation must be set up:

$\small \frac{8}{12}=\frac{10}{x}$ , using $\small x$ as the variable for the missing side.

We then must cross multiply, which leaves us with:

$\small 8x=120$

Lastly, we divide both sides by 8 to solve for the missing side:

$\small \frac{8x}{8}=\frac{120}{8}$

Therefore, the longer sides of the rectangle are each .

The sides of rectangle A measure to , , , and . Rectangle B is similar to Rectangle A. The shorter sides of rectangle B measure each. How long are the longer sides of Rectangle B?

Explanation

In similar rectangles, the ratio of the sides must be equal.

To solve this question, the following equation must be set up:

$\small \frac{8}{12}=\frac{10}{x}$ , using $\small x$ as the variable for the missing side.

We then must cross multiply, which leaves us with:

$\small 8x=120$

Lastly, we divide both sides by 8 to solve for the missing side:

$\small \frac{8x}{8}=\frac{120}{8}$

Therefore, the longer sides of the rectangle are each .

A square has a length of . What must be the length of the diagonal?

$3x\sqrt2$

$3\sqrt2$

$3x^2\sqrt2$

$3\sqrt{2x}$

$9x\sqrt2$

Explanation

A square with a length of indicates that all sides are since a square has 4 equal sides. Use the Pythagorean Theorem to solve for the diagonal.

$c=\sqrt{a^2+b^2} = \sqrt{(3x)^2+(3x)^2}= \sqrt{9x^2+9x^2}= \sqrt{18x^2}= 3x\sqrt2$

A square has a length of . What must be the length of the diagonal?

$3x\sqrt2$

$3\sqrt2$

$3x^2\sqrt2$

$3\sqrt{2x}$

$9x\sqrt2$

Explanation

A square with a length of indicates that all sides are since a square has 4 equal sides. Use the Pythagorean Theorem to solve for the diagonal.

$c=\sqrt{a^2+b^2} = \sqrt{(3x)^2+(3x)^2}= \sqrt{9x^2+9x^2}= \sqrt{18x^2}= 3x\sqrt2$

Suppose a rectangle has side lengths of 7 and 3. Another rectangle has another set of side lengths that are 8 and 4. Are these similar rectangles, and why?

$\textup{No, they are not similar due to proportion.}$

$\textup{No, they are not similar due to length.}$

$\textup{No, they are not similar due to area.}$

$\textup{Yes, dimensions are only one unit more than the lengths of the other rectangle.}$

$\textup{There is not enough information.}$

Explanation

Set up the proportion to determine if the ratios of both rectangles are equal.

If they are, then they are similar.

$\frac{7}{3} =\frac{8}{4}?$

$\frac{7}{3} eq 2$

These are not similar rectangles since their ratios are not the same.

Suppose a rectangle has side lengths of 7 and 3. Another rectangle has another set of side lengths that are 8 and 4. Are these similar rectangles, and why?

$\textup{No, they are not similar due to proportion.}$

$\textup{No, they are not similar due to length.}$

$\textup{No, they are not similar due to area.}$

$\textup{Yes, dimensions are only one unit more than the lengths of the other rectangle.}$

$\textup{There is not enough information.}$

Explanation

Set up the proportion to determine if the ratios of both rectangles are equal.

If they are, then they are similar.

$\frac{7}{3} =\frac{8}{4}?$

$\frac{7}{3} eq 2$

These are not similar rectangles since their ratios are not the same.

The rhombus above is bisected by two diagonals.

If $\angle ADE = x+35$ and $\angle EAB = 2x+10$ then, in degrees, what is the value of the $\angle BCD$ ?

Note: The shape above may not be drawn to scale.

$80^\circ$

$50^\circ$

$15^\circ$

$100^\circ$

$90^\circ$

Explanation

A rhombus is a quadrilateral with two sets of parallel sides as well as equal opposite angles. Since the lines drawn inside the rhombus are diagonals, $\angle DAB,\angle ABC,\angle BCD,$ and $\angle CDA$ are each bisected into two equal angles.

Therefore, $\angle ADE=\angle ABE$ , which creates a triangle in the upper right quadrant of the kite. The sum of angles in a triangle is 180 degreees.

Thus,

$\angle EAB = 2(15)+10=40$

Since $\angle EAB$ is only half of $\angle DAB$ ,

$\angle DAB=2(40)=80$

$\angle DAB=\angle BCD=80$

The rhombus above is bisected by two diagonals.

If $\angle ADE = x+35$ and $\angle EAB = 2x+10$ then, in degrees, what is the value of the $\angle BCD$ ?

Note: The shape above may not be drawn to scale.

$80^\circ$

$50^\circ$

$15^\circ$

$100^\circ$

$90^\circ$

Explanation

Therefore, $\angle ADE=\angle ABE$ , which creates a triangle in the upper right quadrant of the kite. The sum of angles in a triangle is 180 degreees.

Thus,

$\angle EAB = 2(15)+10=40$

Since $\angle EAB$ is only half of $\angle DAB$ ,

$\angle DAB=2(40)=80$

$\angle DAB=\angle BCD=80$

Find the perimeter of the rhombus above.

Explanation

By definition, a rhombus is a quadrilateral with four equal sides whose angles do not all equal 90 degrees. To find the perimeter, we must find the values of x and y. In order to do so, we must set up a system of equations where we set two sides equal to each other. Any two sides can be used to create these systems.

Here is one example:

Eq. 1