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Question Video: Finding the Area of a Triangle Using the Relation between Parallel Lines and Triangles Sharing the Same Base Mathematics • 11th Grade

Given that ๐ด๐ต๐ถ๐ท is a square whose perimeter is 232 cm, find the lengths of line segments ๐ด๐ต and ๐ถ๐ธ. Then find area of โ–ณ๐ธ๐ท๐ต.

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Video Transcript

Given that ๐ด๐ต๐ถ๐ท is a square whose perimeter is 232 centimeters, find the lengths of line segment ๐ด๐ต and line segment ๐ถ๐ธ. Then, find the area of triangle ๐ธ๐ท๐ต.

So here we have our square, which has a perimeter of 232 centimeters. We can recall that the perimeter of a shape is the sum of the outside lengths. So if our square had a length of ๐‘ฅ, we would find the perimeter by adding ๐‘ฅ plus ๐‘ฅ plus ๐‘ฅ plus ๐‘ฅ or working out four times ๐‘ฅ. To find ๐‘ฅ, we take our perimeter of 232 and divide by four, giving us 58 centimeters. So now, we know that the length of every edge in the square is 58 centimeters. And so, for our first question to find the length of line segment ๐ด๐ต, we can write that ๐ด๐ต is 58 centimeters.

The next length that we need to find is that of ๐ถ๐ธ. We can see from our diagram that the lengths ๐ถ๐ธ and ๐ธ๐ต are marked as equal, which means that we could divide our length ๐ถ๐ต of 58 centimeters by two. And since 58 divided by two is 29, then we can say that ๐ถ๐ธ is 29 centimeters. Our final part of the question is to find the area of triangle ๐ธ๐ต๐ท, which we can mark in pink on the diagram. To do this, we need to use the formula that the area of a triangle is half times the base times the height. And itโ€™s important to note that the height here refers to the perpendicular height of the triangle.

So working out the area of triangle ๐ธ๐ท๐ต using the formula, we can take ๐ธ๐ต to be the base of the triangle, which has a length of 29 centimeters. Our perpendicular height will be equivalent to line ๐ถ๐ท, which is 58. And so we have the calculation a half times 29 times 58. We can simplify this calculation by noticing we have the even number 58, which we can easily take a half of, making a calculation of 29 times 29. So our answer is 841 square centimeters.

Therefore, our final answer for all parts is ๐ด๐ต equals 58 centimeters. ๐ถ๐ธ equals 29 centimeters. And the area of triangle ๐ธ๐ท๐ต equals 841 square centimeters.

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