NYS Geometry Regents Exam Practice: Questions 28 & 29

Welcome back to our series on NYS Geometry Regents Exam practice! In this video, we'll tackle questions 28 and 29 from a past exam. These questions are a great way to test your understanding of key geometry concepts, so let's dive in.

Question 28: Proving Triangle Similarity

**The Problem:**

In the diagram below, △ABC is similar to △DEF. Find the length of side DF.

**Solution:**

Since the triangles are similar, their corresponding sides are proportional. This means:

AB/DE = BC/EF = AC/DF

We are given the lengths of AB, DE, BC, and EF. We need to find DF. Let's plug in the values:

6/9 = 8/12 = AC/DF

Simplifying the first two ratios, we get:

2/3 = 2/3 = AC/DF

Therefore, AC/DF = 2/3. To solve for DF, we can cross-multiply:

2DF = 3AC

We are given that AC = 10, so:

2DF = 3(10)

2DF = 30

DF = 30/2

DF = 15

**Answer:** The length of side DF is 15.

Question 29: Finding the Area of a Triangle

**The Problem:**

The vertices of triangle ABC are A(2,1), B(8,1), and C(5,7). Find the area of triangle ABC.

**Solution:**

We can find the area of a triangle using the formula:

Area = (1/2) * base * height

Let's consider side AB as the base of the triangle. The length of AB is 8 - 2 = 6 units. The height of the triangle is the perpendicular distance from point C to line AB.

Since AB is horizontal, the height is simply the difference in y-coordinates between C and any point on AB. We can use point A:

Height = 7 - 1 = 6 units

Now, we can plug in the values into the area formula:

Area = (1/2) * 6 * 6

Area = 18 square units

**Answer:** The area of triangle ABC is 18 square units.

Key Takeaways

These two questions demonstrate important concepts in geometry: similarity and area calculation. Remember to:

Understand the properties of similar triangles, including proportional sides.
Know how to calculate the area of a triangle using the base and height.
Practice applying these concepts to different types of problems.

Keep practicing with past Regents exams and other geometry resources. Good luck with your studies!

NYS Geometry Regents Exam Practice: Questions 28 & 29