Math Problem Statement

the smaller seat of the trapezium is 3, the larger seat is 17. one of the sides is 13, the other is 15. find the height

Solution

To calculate the height of the trapezium given the following:

  • Smaller base (aa) = 3
  • Larger base (bb) = 17
  • One non-parallel side (c1c_1) = 13
  • Other non-parallel side (c2c_2) = 15

We can use the formula for the area of a trapezium combined with the Pythagorean theorem for height. Let's break this into steps:


  1. Step 1: Divide the Trapezium
    • A trapezium can be divided into a rectangle and two right triangles by drawing the height (hh) perpendicular to the bases.
    • This results in two horizontal segments (d1d_1 and d2d_2) that make up the difference between bb and aa: ba=173=14.b - a = 17 - 3 = 14.

  1. Step 2: Express Horizontal Lengths
    • The horizontal segments of the triangles (d1d_1 and d2d_2) are such that d1+d2=14d_1 + d_2 = 14.

  1. Step 3: Height Using Pythagoras
    • For the two right triangles: h2+d12=c12,h^2 + d_1^2 = c_1^2, h2+d22=c22.h^2 + d_2^2 = c_2^2.

    • Substituting d2=14d1d_2 = 14 - d_1 into the second equation: h2+(14d1)2=c22.h^2 + (14 - d_1)^2 = c_2^2.


  1. Step 4: Solve for hh
    • Use both equations and solve for hh, the height.

Let me calculate this for you.from sympy import symbols, Eq, solve

Variables

h, d1 = symbols('h d1')

Given values

c1 = 13 # side 1 c2 = 15 # side 2 diff = 14 # difference between larger and smaller base

Equations based on Pythagoras

eq1 = Eq(h2 + d12, c12) # h^2 + d1^2 = c1^2 eq2 = Eq(h2 + (diff - d1)2, c22) # h^2 + (14 - d1)^2 = c2^2

Solve the system of equations

solution = solve((eq1, eq2), (h, d1)) solution[(-12, 5), (12, 5)][(-12, 5), (12, 5)]successThe solution for the height (hh) of the trapezium is h=12h = 12.

The calculations yield two possible values for hh (12-12 and 1212), but since height cannot be negative, the height of the trapezium is 12 units.


Would you like a detailed explanation of the calculation or have other questions? Here are some related questions:

  1. How can the area of this trapezium be calculated?
  2. What is the significance of dividing the trapezium into triangles?
  3. Can we verify the solution using another method?
  4. What other geometric properties of trapeziums are useful for problem-solving?
  5. How can we generalize this approach to any trapezium?

Tip: Always check whether the assumptions about the figure, like perpendicular height, match the geometry provided.

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Math Problem Analysis

Mathematical Concepts

Geometry
Trapezium properties
Pythagorean theorem
Algebra

Formulas

Area of trapezium = 1/2 * (a + b) * h
Pythagorean theorem: h^2 + d^2 = c^2

Theorems

Pythagorean theorem

Suitable Grade Level

Grades 9-11