One SAT OC practice test problem

Let's take a look at the problem:

The official solution is using the slope of line L. However, we can also use similar triangles to solve this problem. 

Official solution

Since AB is perpendicular to AC, it follows that ABC is a right triangle. By the Pythagorean Theorem, AB = SQR (BC^2 - AC^2) = SQR (5^2 - 3^2) = SQR (25 - 9) = 4. The coordinates of point A are (1,0), so the coordinates of C are (4,0), and the coordinates of B are (1,AB) = (1,4). Thus, the slope of line L is (4-0)/(1-4) = -4/3. Hence the equation of line is y=-4/3*x+b, where b is the y-intercept of L. Point C(4,0) is on L, so we can calculate b=16/3. The answer can be gridded as 16/3. 

The correct answer is 16/3 or as its rounded decimal equivalent, 5.33 .

Alternative solution

First, we know by the Pythagorean Theorem, AB = 4. Because AB is perpendicular to AC and OD is perpendicular to AC(OC), we have DOC and BAC is right angle and ODC = ABC. Hence, we have triangle DOC and BAC is similar triangles by AAA rules.

From similar triangles properties, we have AB/OD = AC/OC. AB = 4, AC = 3 and OC = 4. So, we have 4/OD = 3/4. Therefore, OD = 16/3.

Note:

Properties of Similar Triangles

1. Corresponding angles are congruent (same measure)
2. Corresponding sides are all in the same proportion. Above, OD/AB = OC/AC = DC/BC

How to tell if triangles are similar

Any triangle is defined by six measures (three sides, three angles). But you don't need to know all of them to show that two triangles are similar. Various groups of three will do. Triangles are similar if:

1. AAA (angle angle angle) All three pairs of corresponding angles are the same.
2. SSS in same proportion (side side side) All three pairs of corresponding sides are in the same proportion
3. SAS (side angle side) Two pairs of sides in the same proportion and the included angle equal.