Calculate height of isosceles triangle.
Understand the Problem
The question is asking how to calculate the height of an isosceles triangle. The height can be found using the formula that relates the area of the triangle to its base and height or using the Pythagorean theorem depending on the given dimensions.
Answer
The height is given by $ h = \sqrt{a^2 - \left(\frac{b}{2}\right)^2} $
Answer for screen readers
The height of the isosceles triangle is given by the formula: $$ h = \sqrt{a^2 - \left(\frac{b}{2}\right)^2} $$
Steps to Solve
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Identify the known dimensions Determine the base length and the equal side lengths of the isosceles triangle. Let’s say the base is $b$ and the sides are of equal length $a$.
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Find the length of half the base To use the Pythagorean theorem, we need to divide the base by 2. Therefore, half the base is given by: $$ \text{Half base} = \frac{b}{2} $$
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Apply the Pythagorean theorem Using the right triangle formed by the height ($h$), half of the base ($\frac{b}{2}$), and the equal side ($a$), we can express this relationship as follows: $$ a^2 = h^2 + \left(\frac{b}{2}\right)^2 $$
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Rearrange the equation to solve for height To isolate $h^2$, rearrange the equation: $$ h^2 = a^2 - \left(\frac{b}{2}\right)^2 $$
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Take the square root to find height To find the height $h$, take the square root of both sides: $$ h = \sqrt{a^2 - \left(\frac{b}{2}\right)^2} $$
The height of the isosceles triangle is given by the formula: $$ h = \sqrt{a^2 - \left(\frac{b}{2}\right)^2} $$
More Information
This formula shows how the height of an isosceles triangle can be calculated using the lengths of its sides. The Pythagorean theorem is essential in deriving the height from the other dimensions, allowing us to consider the triangle's geometric properties.
Tips
- Forgetting to divide the base by 2: It’s important to correctly calculate half the base when applying the Pythagorean theorem.
- Not using the correct side lengths: Ensure you are using the lengths of the equal sides and the base correctly.
- Mistaking $h$ as a hypotenuse: Remember that in the triangle formed, the height is not the hypotenuse; ensure to represent sides appropriately in the theorem.
