This is not a FOIL problem, as we are adding rather than multiplying the terms in parentheses.

Add like terms together:

 has no like terms.

Combine these terms into one expression to find the answer:

To simplify this expression you need to carefully remove the parentheses by distributing the negative sign across all terms in the second set of parentheses. To do this, just take the opposite of each of those signs. That gives you:

Then you can combine like terms:

To solve this expression, you need to remove the parentheses, being careful to account for the negative prior to the second set of parentheses. To distribute that negative, take the opposite of each sign for that set of values. Then, combine like terms:

A look at the answer choices tells you that you need to simplify the expression to arrive at fewer terms. To simplify this expression, you need to remove the parentheses and combine like terms. Fortunately, the process is to add the two parentheses and there are no coefficients or negatives between them, so you can simply lift the parentheses entirely and then combine.  That leaves you with:

A key to solving this problem comes in recognizing that you’re dealing with similar triangles. Because lines BE, CF, and DG are all parallel, that means that the top triangle ABE is similar to two larger triangles, ACF and ADG. You know that because they all share the same angle A, and then if the horizontal lines are all parallel then the bottom two angles of each triangle will be congruent as well. You’ve established similarity through Angle-Angle-Angle.

This means that the side ratios will be the same for each triangle.  And for the top triangle, ABE, you know that the ratio of the left side (AB) to right side (AE) is 6 to 9, or a ratio of 2 to 3.  Note then that the remainder of the given information provides you the length of the entire right-hand side, line AG, of larger triangle ADG.  If AE is 9, EF is 10, and FG is 11, then side AG is 30.  And since you know that the left-hand side has a 2:3 ratio to the right, then line segment AD must be 20.

This problem tests the concept of similar triangles. First, you should recognize that triangle ACE and triangle BDE are similar. You know this because they each have the same angle measures: they share the angle created at point E and they each have a 90-degree angle, so angle CAE must match angle DBE (the top left angle in each triangle.

Because these triangles are similar, their dimensions will be proportional. Since sides, AC and BD - which are proportional sides since they are both across from the same angle, E - share a 3:2 ratio you know that each side of the smaller triangle (BDE) will be  as long as its counterpart in the larger triangle (ACE). Consequently, if the bottom side CE in the larger triangle measures 30, then the proportional side for the smaller triangle (side DE) will be  as long, measuring 20.

Since the question asks for the length of CD, you can take side CE (30) and subtract DE (20) to get the correct answer, 10.

This problem hinges on your ability to recognize two important themes: one, that triangle ABC is a special right triangle, a 6-8-10 side ratio, allowing you to plug in 8 for side AB. And secondly, triangles ABC and CDE are similar triangles. You know this because each triangle is marked as a right triangle and angles ACB and ECD are vertical angles, meaning that they’re congruent. Since all angles in a triangle must sum to 180, if two angles are the same then the third has to be, too, so you’ve got similar triangles here.

With that knowledge, you know that triangle ECD follows a 3-4-5 ratio (the simplified version of 6-8-10), so if the side opposite angle C in ABC is 8 and in CDE is 12, then you know you have a 9-12-15 triangle. With the knowledge that side CE measures 15, you can add that to side BC which is 10, and you have the answer of 25.

An important point of recognition on this problem is that triangles JXZ and KYZ are similar. Each has a right angle and each shares the angle at point Z, so the third angles (XJZ and YKZ, each in the upper left corner of its triangle) must be the same, too.

Given that, if you know that JX measures 16 and KY measures 8, you know that each side of the larger triangle measures twice the length of its counterpart in the smaller triangle.

You also have enough information to solve for side XZ, since you’re given the area of triangle JXZ and a line, JX, that could serve as its height (remember, to use the  base x height equation for the area of a triangle, you need base and height to be perpendicular; lines JX and XZ are perpendicular). Since , you can see that XZ must measure 10. And since XZ will be twice the length of YZ by the similarity ratio, YZ = 5, meaning that XY must also be 5.

The first important thing to note on this problem is that for each triangle, you’re given two angles: a right angle, and one other angle. Because all angles in a triangle must sum to 180 degrees, this means that you can solve for the missing angles.

In ABC, you have angles 36 and 90, meaning that to sum to 180 the missing angle ACB must be 54. And in XYZ, you have angles 90 and 54, meaning that the missing angle XZY must be 36.

Next, you can note that both triangles have the same angles: 36, 54, and 90.  This means that the triangles are similar, which also means that their side ratios will be the same. You just need to make sure that you’re matching up sides based on the angles that they’re across from.

You’re given the ratio of AC to BC, which in triangle ABC is the ratio of the side opposite the right angle (AC) to the side opposite the 54-degree angle (BC). In triangle XYZ, those sides are XZ and XY, so the ratio you’re looking for is .

Because the triangles are similar, you can tell that if the hypotenuse of the larger triangle is 15 and the hypotenuse of the smaller triangle is 10, then the sides have a ratio of 3:2 between the triangles.  This allows you to fill in the sides of XYZ: side XY is 6 (which is 2/3 of its counterpart side AB which is 9) and since YZ is 8 (which is 2/3 of its counterpart side, BC, which is 12).

Since the area of a triangle is  Base * Height, if you know that you have a base of 8 and a height of 6, that means that the area is .