The area of the triangle is 35 square meters. Find the dimensions of the triangle given the picture shown.
Understand the Problem
The question provides the area of a triangle as 35 square meters. The base is represented as (g - 8) meters and the height as (g - 11) meters. The goal is to find the values of the base and height by determining the value of 'g'. We can use the formula for the area of a triangle (Area = 1/2 * base * height) to set up an equation and solve for 'g'. Once 'g' is found, we can substitute it back into the expressions for the base and height to find their respective values.
Answer
Base: $10$ m Height: $7$ m
Answer for screen readers
Base: $10$ meters Height: $7$ meters
Steps to Solve
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Write the formula for the area of a triangle The area of a triangle is given by: $$ Area = \frac{1}{2} \cdot base \cdot height $$
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Substitute the given values into the formula We are given the area as 35, the base as $(g - 8)$, and the height as $(g - 11)$. Substituting these values into the formula: $$ 35 = \frac{1}{2} (g - 8)(g - 11) $$
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Simplify the equation Multiply both sides by 2: $$ 70 = (g - 8)(g - 11) $$ Expand the right side: $$ 70 = g^2 - 11g - 8g + 88 $$ $$ 70 = g^2 - 19g + 88 $$
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Rearrange the equation into a quadratic equation Subtract 70 from both sides: $$ 0 = g^2 - 19g + 18 $$
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Solve the quadratic equation We need to find two numbers that multiply to 18 and add up to -19. These numbers are -1 and -18. $$ 0 = (g - 1)(g - 18) $$ So, $g = 1$ or $g = 18$.
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Check the solutions If $g = 1$, then the base is $1 - 8 = -7$ and the height is $1 - 11 = -10$. Since the dimensions of a triangle cannot be negative, $g = 1$ is not a valid solution.
If $g = 18$, then the base is $18 - 8 = 10$ and the height is $18 - 11 = 7$. These are both positive values, so $g = 18$ is a valid solution.
- Find the dimensions of the triangle Base = $g - 8 = 18 - 8 = 10$ meters Height = $g - 11 = 18 - 11 = 7$ meters
Base: $10$ meters Height: $7$ meters
More Information
The value of $g$ is $18$. The dimensions of the triangle are a base of 10 meters and a height of 7 meters. The area can be verified by substituting the dimensions into the area formula: $Area = \frac{1}{2} * 10 * 7 = 35$ square meters.
Tips
A common mistake is to forget to check the validity of the solutions for $g$. In this case, one solution led to negative dimensions, which are not physically possible, so it must be discarded.
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