Math Problem Statement

Find the length and width of the rectangle. Solve for the unknown angle measures in the given triangles.

Solution

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Problem 6: Rectangle Dimensions

Given:

  • The perimeter of the rectangle GHGH is 170 feet.
  • Length GH=4xGH = 4x, Width HI=6x−15HI = 6x - 15.

Perimeter Formula: P=2×(Length+Width)P = 2 \times (\text{Length} + \text{Width})

Substitute the given values: 170=2×(4x+(6x−15))170 = 2 \times (4x + (6x - 15))

Simplify and solve for xx:

170 = 20x - 30 \\ 200 = 20x \\ x = 10$$ **Dimensions:** - Length: $$4x = 4 \times 10 = 40$$ feet - Width: $$6x - 15 = 6 \times 10 - 15 = 45$$ feet --- ### Problem 5: Triangle $$\triangle LMN$$ **Given:** - $$\angle L$$: $$2x + 10^\circ$$ - $$\angle M$$: $$4x^\circ$$ - $$\angle N$$: $$90^\circ$$ **Triangle Angle Sum Theorem:** $$\angle L + \angle M + \angle N = 180^\circ$$ Substitute the values: $$(2x + 10) + 4x + 90 = 180 \\ 6x + 100 = 180 \\ 6x = 80 \\ x = \frac{80}{6} = \frac{40}{3} \approx 13.33$$ **Angles:** - $$\angle L = 2x + 10 = 2 \times \frac{40}{3} + 10 = \frac{80}{3} + \frac{30}{3} = \frac{110}{3} \approx 36.67^\circ$$ - $$\angle M = 4x = 4 \times \frac{40}{3} = \frac{160}{3} \approx 53.33^\circ$$ - $$\angle N = 90^\circ$$ --- ### Problem 4: Triangle $$\triangle DEF$$ **Given:** - Side lengths are provided. Use the Law of Cosines or Sin Rule to calculate angles. Would you like me to calculate this explicitly for $$\triangle DEF$$? Let me know. I can also provide detailed step-by-step work for all solutions. --- Would you like further clarification or details? Here are follow-up questions you might consider: 1. How do we derive the perimeter formula for a rectangle? 2. What happens when the given angles in a triangle don't add up to 180°? 3. How can you solve a triangle without knowing all angles? 4. Can we use trigonometry to verify triangle properties? 5. How do units affect solutions in these problems? **Tip:** Always recheck your calculations, especially in multi-step algebraic problems!

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Math Problem Analysis

Mathematical Concepts

Algebra
Geometry
Triangle Angle Sum Theorem
Perimeter of a Rectangle
Law of Cosines (if applied to triangle DEF)

Formulas

P = 2(Length + Width) for perimeter of a rectangle
Triangle Angle Sum Theorem: sum of angles in a triangle is 180°
Law of Cosines: c^2 = a^2 + b^2 - 2ab*cos(C) (if relevant to triangle DEF)

Theorems

Triangle Angle Sum Theorem

Suitable Grade Level

Grades 8-10

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