Math Problem Statement

Given two similar triangles, ABC and DEF, find the value of x and y.

Solution

The two triangles ABC\triangle ABC and DEF\triangle DEF are given as similar (ABCDEF\triangle ABC \sim \triangle DEF). From the properties of similar triangles, we know that corresponding sides are proportional.

The corresponding sides of the triangles are:

  • ABAB corresponds to DEDE
  • ACAC corresponds to DFDF
  • BCBC corresponds to EFEF

We can now set up ratios of corresponding sides to solve for xx and yy.

Step 1: Set up the proportion for side ACAC and DFDF

Given:

  • AC=4y2AC = 4y - 2
  • DF=12DF = 12

Using the ratio ACDF=ABDE\frac{AC}{DF} = \frac{AB}{DE}, we can use corresponding sides: 4y212=2114\frac{4y - 2}{12} = \frac{21}{14} Simplify 2114\frac{21}{14} to 32\frac{3}{2}: 4y212=32\frac{4y - 2}{12} = \frac{3}{2} Cross-multiply: 2(4y2)=3×122(4y - 2) = 3 \times 12 Simplify: 8y4=368y - 4 = 36 Solve for yy: 8y=36+48y = 36 + 4 8y=408y = 40 y=408=5y = \frac{40}{8} = 5

Step 2: Set up the proportion for side ABAB and DEDE

Now, to solve for xx, we use the ratio ABDE=BCEF\frac{AB}{DE} = \frac{BC}{EF}.

Given:

  • AB=21AB = 21
  • DE=14DE = 14
  • BC=24BC = 24
  • EF=16EF = 16

Use the ratio: ABDE=BCEF\frac{AB}{DE} = \frac{BC}{EF} Simplify ABDE=2114=32\frac{AB}{DE} = \frac{21}{14} = \frac{3}{2}: 32=2416\frac{3}{2} = \frac{24}{16} Since the ratio holds, the corresponding sides are proportional.

For side AD=xAD = x, we know that: x=16(as corresponding side).x = 16 \, \text{(as corresponding side)}.

Final Answers:

  • y=5y = 5
  • x=16x = 16

Would you like further details or explanations?


Here are some related questions:

  1. What is the relationship between the angles in similar triangles?
  2. How do you calculate the area of similar triangles if one side length is known?
  3. How does the ratio of corresponding sides affect the perimeter of similar triangles?
  4. What happens to the ratio of the areas when two triangles are similar?
  5. Can two triangles be similar if their corresponding angles are not equal?

Tip: In similar triangles, the ratio of any two corresponding sides is always constant.

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Math Problem Analysis

Mathematical Concepts

Similar Triangles
Proportionality
Algebra

Formulas

Ratio of corresponding sides: (AC / DF) = (AB / DE)
Cross multiplication for solving proportions

Theorems

Theorem of Similar Triangles
Proportionality of corresponding sides

Suitable Grade Level

Grades 8-10