Math Problem Statement

In Figure, ΔABE is similar to ΔACD. What is the value of x?

Solution

The image presents two triangles, ABE\triangle ABE and ACD\triangle ACD, where it is stated that these triangles are similar. To find the value of xx, let's use properties of similar triangles.

Step-by-step explanation:

Step 1: Identify corresponding sides

Since ABEACD\triangle ABE \sim \triangle ACD, the ratios of corresponding sides will be equal. The corresponding sides are:

  • ABAB and ACAC
  • BEBE and CDCD
  • AEAE and ADAD

Step 2: Write the proportions of corresponding sides

Using the property of similar triangles, we set up the following proportion:

ABAC=BECD=AEAD\frac{AB}{AC} = \frac{BE}{CD} = \frac{AE}{AD}

We are given the lengths:

  • AB=8AB = 8
  • AC=8+3=11AC = 8 + 3 = 11 (since BC=3BC = 3)
  • AE=10AE = 10
  • AD=10+xAD = 10 + x (since ED=xED = x)

Step 3: Set up the ratio for AEAE and ADAD

We will use the third ratio for this calculation: AEAD=ABAC\frac{AE}{AD} = \frac{AB}{AC}

Substitute the known values: 1010+x=811\frac{10}{10 + x} = \frac{8}{11}

Step 4: Solve for xx

Now, cross-multiply to solve for xx: 10×11=8×(10+x)10 \times 11 = 8 \times (10 + x) 110=80+8x110 = 80 + 8x 11080=8x110 - 80 = 8x 30=8x30 = 8x x=308=3.75x = \frac{30}{8} = 3.75

Thus, the value of xx is 3.75.

Would you like more details or have any questions?


Here are 5 related questions to expand on this problem:

  1. What are the basic properties of similar triangles?
  2. How can you verify that two triangles are similar using angle relationships?
  3. What are other methods to solve for unknowns in similar triangles?
  4. Can the side lengths in similar triangles be in different proportions for different sides?
  5. How would the problem change if the triangles were not similar?

Tip: When working with similar triangles, always double-check that you've correctly matched corresponding sides before setting up proportions.

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Math Problem Analysis

Mathematical Concepts

Similar Triangles
Proportionality
Algebra

Formulas

Ratio of corresponding sides in similar triangles: AB/AC = BE/CD = AE/AD
Proportional equations involving algebraic manipulation

Theorems

Similarity Theorem for Triangles

Suitable Grade Level

Grades 8-10