Math Problem Statement

Find the missing side length in two similar triangles ΔABC and ΔPQR, given: AB = 5, AC = 6, PQ = 20, PR = 24, QR = 32, and BC = ?.

Solution

To solve for the missing side length BCBC in the two similar triangles ABC\triangle ABC and PQR\triangle PQR, we will use the property of similar triangles: corresponding sides are proportional.

Step 1: Identify the corresponding sides

From the given triangles:

  • ABAB corresponds to PQPQ,
  • ACAC corresponds to PRPR,
  • BCBC corresponds to QRQR.

The lengths are:

  • AB=5AB = 5, PQ=20PQ = 20,
  • AC=6AC = 6, PR=24PR = 24,
  • QR=32QR = 32, BC=?BC = ?.

Step 2: Set up the proportion

Using the corresponding sides ABPQ=BCQR\frac{AB}{PQ} = \frac{BC}{QR}: 520=BC32\frac{5}{20} = \frac{BC}{32}

Step 3: Solve for BCBC

Simplify 520\frac{5}{20} to 14\frac{1}{4}: 14=BC32\frac{1}{4} = \frac{BC}{32}

Now, solve for BCBC by cross-multiplying: 132=4BC1 \cdot 32 = 4 \cdot BC BC=324=8BC = \frac{32}{4} = 8

Final Answer:

The missing side BCBC is 8.


Would you like me to explain any step further or solve another problem?

Related Questions:

  1. How do you prove two triangles are similar?
  2. What are the properties of similar triangles?
  3. How do you solve for an unknown side in similar triangles?
  4. How do you scale triangles using a ratio?
  5. Can the similarity ratio of triangles be used in real-life applications?

Tip:

Always ensure you set up proportions carefully when solving problems involving similar triangles!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Similar Triangles
Proportionality
Cross-Multiplication

Formulas

Corresponding sides in similar triangles are proportional: AB/PQ = BC/QR

Theorems

Triangle Similarity Theorem
Proportionality Theorem

Suitable Grade Level

Grade 8-10