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 () = 3
- Larger base () = 17
- One non-parallel side () = 13
- Other non-parallel side () = 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:
- Step 1: Divide the Trapezium
- A trapezium can be divided into a rectangle and two right triangles by drawing the height () perpendicular to the bases.
- This results in two horizontal segments ( and ) that make up the difference between and :
- Step 2: Express Horizontal Lengths
- The horizontal segments of the triangles ( and ) are such that .
- Step 3: Height Using Pythagoras
-
For the two right triangles:
-
Substituting into the second equation:
-
- Step 4: Solve for
- Use both equations and solve for , 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 () of the trapezium is .
The calculations yield two possible values for ( and ), 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:
- How can the area of this trapezium be calculated?
- What is the significance of dividing the trapezium into triangles?
- Can we verify the solution using another method?
- What other geometric properties of trapeziums are useful for problem-solving?
- 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
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