Math Problem Statement
Solution
The problem asks for the height of a smaller triangle given that the ratio of the areas of two triangles with equal bases is 8:5, and the height of the larger triangle is 12 cm.
We can use the formula for the area of a triangle:
Since the bases of the triangles are equal, the ratio of their areas depends only on the heights of the triangles. Let the height of the smaller triangle be .
The ratio of the areas of the triangles is:
This simplifies to the ratio of the heights:
Now, solve for :
Thus, the height of the smaller triangle is 7.5 cm.
Would you like further details or have any questions? Here are five related questions to expand on this problem:
- How would the ratio change if the bases of the triangles were different?
- Can the ratio of the heights be directly used if both base and height change?
- How would you solve this problem if the ratio of areas was given for a trapezoid instead of a triangle?
- If the height of the smaller triangle was known, how could you calculate the height of the larger triangle?
- How does the formula for the area of a triangle help in comparing different properties of triangles?
Tip: When the bases are equal, comparing areas is the same as comparing heights. This makes calculations much simpler!
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Math Problem Analysis
Mathematical Concepts
Ratio of Areas
Triangles
Proportions
Formulas
Area of a triangle = (1/2) * base * height
Theorems
The ratio of the areas of two triangles with equal bases is proportional to their heights.
Suitable Grade Level
Grades 8-10