Video: Finding the Area of a Triangle given Its Base and Height as Algebraic Terms

Given that ๐ด๐ท = (๐‘ฅโด๐‘ฆยณ)ยฒ cm and ๐ถ๐ต = (6๐‘ฅ๐‘ฆโด)ยฒ cm, determine, in its simplest form, the algebraic expression that represents the area of triangle ๐ด๐ต๐ถ.

03:19

Video Transcript

Given that ๐ด๐ท equals ๐‘ฅ to the fourth ๐‘ฆ cubed squared centimeters and ๐ถ๐ต equals six ๐‘ฅ๐‘ฆ to the fourth squared centimeters, determine in its simplest form the algebraic expression that represents the area of the triangle ๐ด๐ต๐ถ.

So it seems like we have a lot going on here; letโ€™s break it down. Weโ€™re given the length of ๐ด๐ท, which is the height of our triangle. They tell us that that equals ๐‘ฅ to the fourth ๐‘ฆ cubed squared. And then weโ€™re given the distance of ๐ถ๐ต, which is the base of our triangle, a height of ๐‘ฅ to the fourth ๐‘ฆ cubed squared and a base of six ๐‘ฅ๐‘ฆ to the fourth squared.

Before we find the area, letโ€™s simplify both the height and the base. To do that, I need to distribute the exponents, the powers, to the powers. And we remember our power to a power rule that says if weโ€™re taking the power of a power, we multiply the exponents. And thatโ€™s what weโ€™ll do here; weโ€™ll multiply four times two to find the power of our ๐‘ฅ; and weโ€™ll multiply three times two to find the power of our ๐‘ฆ.

So simplified, our height can be written as ๐‘ฅ to the eighth power times ๐‘ฆ to the sixth power. Weโ€™ll also need to distribute our powers to simplify the base. In this case, weโ€™ll get six squared times ๐‘ฅ squared, and then weโ€™ll multiply four times two to find the exponent of ๐‘ฆ. And the simplified form of our base becomes 36๐‘ฅ squared ๐‘ฆ to the eighth power.

And we wanna find the area of this triangle. The formula for finding the area of a triangle equals one half height times the base. Our area will be one half times ๐‘ฅ to the eighth ๐‘ฆ to the sixth times 36๐‘ฅ squared ๐‘ฆ to the eighth.

What will do now is combine our like terms: one half times 36 equals 18; ๐‘ฅ to the eighth power times ๐‘ฅ squared equals ๐‘ฅ to the tenth power โ€” here we add our exponents because weโ€™re multiplying two powers with the same base โ€” and finally ๐‘ฆ to the sixth power times ๐‘ฆ to the eighth power.

This is ๐‘ฆ to the 14th power. When we multiply a power times a power and they have the same base, we add the exponents. So we added six plus eight to equal 14. The area of this triangle would be equal to 18๐‘ฅ to the tenth ๐‘ฆ to the 14th centimeters squared.

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